Lemons HQ will want to see insurance and registration documents at the start of the rally (also a valid driving license), so if you do buy something at the last minute make sure you have real documents with you. You probably can, for example, arrange insurance with just a phone call but that won't necessarily give you something sufficiently convincing to show HQ. A quick in-person visit to an insurance agent's office will. I understand there are some fancy, newfangled methods of displaying proof of insurance via smart phone but I have no idea how HQ would feel about that. Since you're coming from Canada it's probably worth calling a few US insurance firms well in advance to ask about their policies for foreign nationals just in case it takes extra time to get things going or if there's something specific you need to do. By the way, if you mention the word "rally" during the conversation, be warned that many insurance agents will immediately assume that means "illegal street racing and/or mass murder" and will not want to talk with you. Once they've reached that conclusion you won't be able to change their mind.
Similiarly, although a cop might let you slide with just a recent bill of sale and expired plates, HQ won't. You'll need actual registration. It can certainly be temporary registration, which is to say no current license plate but just a sheet of paper taped in the back window or however the state in question does things, but it does have to be current, real registration.
Having said that, the rally folks love to see people buy and drive, or attempt to drive, something entertaining, sketchy, or both, at the last minute, so go for it.
1982 MG Metro 1300: IOE 2015 Pacific Northworst GP, Longest Distance 2010 Cd'L Box Wine Country Classic
1980 KV Mini 1: Worst of Show and Fright Pig Supremo 2009 Concours d'Lemons
1978 H Special: Second-Round Elimination 2010 Lemons Pinewood Derby at Sears Pointless
1967 SAAB 96: IOE 2012 Pacific Northworst GP