Any food grade container will work. Lots of guys get buckets from bakeries that formerly held frosting, etc. You'll also see them on Craigslist pretty often. Usually about a dollar a bucket. You don't need to cover them, but it is nice. The old style taps usually have a hook to hang a bucket from, they're expensive if you buy them new though. Might luck out on Craigslist. If you hang buckets on the trees, you don't need a lid, but it's nice. Keeps the rain and squirrels out. With no lids you gotta dump buckets when it rains or when you find critters in them.
I see a lot of guys use the new plastic taps (which only cost $.25) and a few feet of sap tubing to reach a bucket on the ground. In that situation a lid is pretty nice, you can drill a hole for the line, so it fits tight and doesn't get knocked out. I've even seen guys do that with milk jugs, though you don't have much capacity that way. If your trees are decent size (22" diameter plus) you can put two taps running into one bucket.
I don't know when you're thinking of tapping, but I've got a bunch of old plastic taps with a couple feet of tubing already on them. It's garbage to me, so if you pay shipping they're all yours. Should have a couple dozen at least.