From a building supply place, get a piece of at least 3/4" plywood (possibly oriented strand board) - check what size you need in order to accomodate whatever number of stacks that you want. And the storage space which they'll occupy when not in use. You'll need a little extra space between the stacks, in order to avoid the need for a shoehorn. Also a little extra around the edges to accomodate a guardrail.

Get four heavy duty rubber or plastic rollers, larger size (likely about 4" or larger caster) that can carry as many hundreds of pounds as the number of chairs that you want to stack weigh, plus some extra for the cart, plus some extra so that they don't wear out soon. You'll want them with a flat fastening surface, in order to screw them to the four corners of the piece of plywood.

You'll want at least two of the casters set up so that they swivel (unless you want the cart to run forever along one straight line) - or have a horse, even an elephant, to drag one end of the cart (hard on the casters) so that it will travel in another direction - which it will want to keep doing forever. You'll want the swivel to operate with roller bearings, or they'll be difficult to get to turn. Install the two swivelling casters at one end of the cart.

Some like to have four swivelling casters, but such carts tend to have a mind of their own and frequently prefer to go in a direction other than the one chosen by the operator (which may lead to some skinned knuckles, even profanity).

Get some pieces of 1" x 2" board and fasten them with screws along the sides and ends of the upper side of the cart, in order that the stacks of chairs don't fall off.

You may want to get some pieces of, say 3/4" galvanized water pipe (likely from scrap metal dealer, plumbing contractor, etc.), with four 90 degree nipples, to make a push handle. A horizontal bar slightly shorter than the width of the cart, enough shorter that the bars going under the cart don't interfere with the casters under the cart - including room for them to swivel, if you install the swivelling casters at the handle end of the cart. Screw a nipple on each end, then two vertical bars to go down to the cart level, with another nipple at the bottom end, then short pieces of pipe to go under the cart, with holes bored in them and the bottom of the cart to bolt them to the cart. It would be nice if you could add a brace to the handle, but if you do, it will likely interfere with the loading of the stacks of chairs at that end of the cart.

On the other hand, you can get about 8' of about 1/2 - 3/4" rope, drill 2 holes near one end the cart, about a quarter of the way in from the edge, push one end of each rope through the hole from the top side, and tie a knot in the end of the rope, under the cart. You'll have much less control using this method than if you can have a stay-in-place handle.

I don't have a suggestion as to whether it would be better to install the swivelling casters at the front end or the handle-end of the cart. Possibly the handle end - then you can aim the front where you want it to go, and move the castered end, under the handle, where you want it to go, more easily.

If you need further explanation, give us a shout.

ole joyful