It's definitely possible to tow a team trailer with a minivan - if the trailer isn't too big. I've seen team trailers that are huge, and ones that aren't.
The real question is what are your requirements? There are typical sizes for trailers:
Using this as a guide, collect together all of the stuff you want in your pit, get a nice open area, and tape out the dimensions. Then you can try arranging things and see what size you need to accommodate your stuff, then do some more detailed investigation into trailers around that size.
When looking at features, consider loading and unloading. Some trailers have a door that drops down to form a ramp, which makes loading and unloading a little easier, while others open like doors, forcing you to either bring a separate ramp or lift everything into the trailer. For my team's trailer, we made sure to get the ramp kind, and then added a couple of small sections of diamond plate on hinges to help bridge the gaps and make wheeling things up easier. It works really well!
Also consider lighting - you'll want some light in there. You can do that two ways - trailers can come with built in dome lights, wired up to the trailer connection to provide power, or you can get some relatively cheap, battery powered LED lights and stick them in places. You've probably seen the infomercials for such products
