I am a small person without access to power tools and didn't have much help for the project, so anything that heavy was out of the question (and the puzzle is huge). Otherwise I would have quite possibly done this!

Plywood is really good. MDF is also an option. But for the size of 9.000 pieces even some sturdy paper (old posters or calendar sheets) can be good enough as a base. With good glue (e.g. some latex binder) the jigsaw puzzle will already be quite stable on its own.

Having a soft base helps when transporting. I once had to move a 9.000 piece puzzle - and without some bending it would have been difficult getting it into the car.

For the biggest jigsaws out there it gets difficult to find a good wall for it. Weight and bending of wood or MDF can become an issue. And even some walls are not as flat as they look like...

For stability I would go with an expensive multi-ply 1/4" plywood. Not your big-box plywood, I'm talking the high-end cabinet-maker type of plywood (Euro-ply, Apple Ply, etc.).

Honestly, if you aren't planning on making a regular thing of it, MDF is probably a better choice. It's more stable than even multi-ply, and cheaper too. The ply wins on edge and corner durability though. On the third hand, MDF from e.g. Home Depot is pretty crap. If you're already going to a better lumberyard, maybe the ply is worth the additional dollars now that you've gone to the trouble of finding a better lumberyard.

Further considerations:

Multi-ply is frequently available in 5'x5' sheets and can be tough to find in 4'x8's. MDF is universally 49"x97" (so you can net a 4'x8' after accounting for the dinged up edges). The size of your puzzle may inform your choice.

Home Depot et al. will usually cut a sheet to size on their panel saw. Your local lumberyard may or may not be interested in doing so for a small order.

Home Depot et al. generally aren't interested in delivering for an even half reasonable price if they do it at all. A real lumberyard might be willing to do so, though it's unlikely to be cheap for a single sheet order ($25 for an order under $500, last I needed a delivery. Cheaper than owning a truck or van for me). They also usually dispatch trucks on a schedule. Mine goes north on Tuesdays and south on Thursdays, for example. Plan ahead and don't expect your stuff to show up like Amazon packages the next day.

MDF was also my choice for my largest gluing project. I needed about 7m² for the 24.000 Educa. MDF is cheap and easily available in big sizes. I'd use Plywood for things <1m², it looks nice.

For the 9.000s I always used paper for the backside (old calendars are a good source for big sheets with excellent quality). Using one wooden lath at the top was enough to ensure stability for a 18.000. The 9.000s didn't need that - just paper and a bit of duct-tape on the backside were enough.

You can do a puzzle on any flat surface, but there's always the problem of how to glue the finished puzzle to that flat surface without turning the whole thing upside down. After all, it's the backside of the puzzle that you want the glue to be on.

I found that a coating of quick-drying, low-viscosity liquid glue on the front side works pretty well, because the glue easily flows into the gaps between the pieces, and then into the space between the puzzle and the board. Once the glue dries off, all the pieces and the backing board become a single chunk of cellulose-infused plastic. This is probably much better than either the thick mod podge or the adhesive sheet that OP tried to use at first.

My strategy was to buy two plywood boards, and put the puzzle in between those to be able to turn it upside down so I could glue it.

I was thinking take it apart in sections, spread glue on each section's area one by one, putting them back in place as you go. The pieces stay the right way up the whole time.

Anything over about 1/4" thick plywood is going to be heavy at the size of that puzzle. At the same time, thinner material (and I'm thinking of Masonite) is not going to be very stable — could end up bowing in the center.

I was thinking the same, the only issue would be cutting the sheet to the exact size which requires a table saw that, I suppose, she had no access to (I guess, a circular saw with another large plywood for a guide would do as well). Perhaps, it could be ordered, too?

Or use a jigsaw. The cutting tool, not the cardboard pieces.

Or sheet of hardboard and a stanley knife, which is probably the most sensible option in context.

Jigsaws make rather poor cuts (can use router afterwards but that's significantly more involved than a circular saw). Personally, I'd not use the jigsaw for pieces I'd hang on the wall, esp. not use jigsaw free hand. On hardboard - no direct experience with, yet I'd expect it doesn't like moisture, so it has to be properly sealed.

Jigsaw cuts look like garbage IME. The boxcutter knife would work, I've done that to trim plywood before and while it takes a while the cut is sharp and precise.

Better blades go a long way towards not splintering your workpiece. The Bosch ones are excellent if you get the "clean cuts in clean wood" ones. Note that there are two incompatible shank styles: T and U. Get the ones that work with your saw.

For sure. Couple other issues, one, the blade tends to wander, leaving not-perfectly-straight sections. You can mitigate by going slow and careful. Second, it needs a straight edge guide offset to the blade, which is harder to set up than direct.

All in all when I needed a one-off cut I used a box cutter knife, and when I needed a series I used a circular saw with a plywood blade. The setup is similar (needs an offset guide) but the cut is much straighter, faster, and less error prone. I think the only time I'd use a jigsaw is for making non-straight cuts like some curved artistic shape or something.

Rail saw would do it easily too. The framing place would probably have something suitable.

If by rail saw you mean a track saw, I heartily agree. Many people who have no room for (or are intimidated by) a table saw should look into purchasing a track saw. (Still a spinning, sharp blade though and not to be trifled with.)

I would not have enough room to install a proper track saw but circular one is mobile and can even cut on the floor with polystyrene foam below. Well, technically I can install the guide rail on the floor in the same fashion.

(Like mentioned I did have it in the reply as well)

The track saws that I am describing are in fact simply a circular saw + guide rail. And cutting on the floor with foam below is exactly the way to get into it.

Proper track saws though tend to come with finer tooth blades with a thinner kerf so that they cut cleaner than a circular saw. They also retract within a kind of housing so that they're safer — only plunging down when you lean on them to make the cut.

Once you use one, you'll not go back to your circular saw unless, for example, you're trying to cut some framing lumber in place.

I have a Makita [1] which I like a lot, but my sister, wanting to save a little dosh, got a Kreg [2] and is happy with it.

[1] https://www.amazon.com/Makita-SP6000J1-Plunge-Circular-Stack...

[2] https://www.amazon.com/Kreg-ACS-SAWBB-Plunge-Cut-replaces-AC...

I did have the plunge saw (another name) as an option but then removed in favor of a easier solution of circular saw + plywood.

Pretty sure Mod Podge is just expensive PVA.