Oooh, did you guys have a nice time? Like it?
We had a compact rental, no issues whatsoever, and we flew into Tetons for 2 days, then up through to Yellowstone Lake then Canyon, to Lamar Valley, then back down to Old Faithful, Mammoth, and out through Bozeman. Didn't need an SUV for any of it.
If you are entering North entrance, there are a bunch of stores and shops right before. Get some huckleberry ice cream on your way in or out, it's so good!!
One of the best things I got was a wilderness guide... it's like a laminated sheet front and back with pictures of the most common animals, and cost $5 two years ago. (ETA: Just checked mine, it’s called “Macs Field Guide to Yellowstone and Grand Teton National Parks”.) I got it at our second hike in Tetons actually, and then used it the rest of our trip religiously. Best thing I bought!! I loved having it to identify birds, snakes, small mammals, etc. So if you have kids, I'm sure they would love that! (I am a 40 yo woman and loved it, LOL. But I did see it first from a kid using it hiking). We saw it in most visitor center gift shops.
Have a blast! We also went the week before memorial day and thought it was the best time... would go again at this time for sure!