How strange is it (as an undergrad) to email a professor from another institution about possibly working in their lab?
Not strange at all, just don't expect a quick response! 

A plan of attack can be summarised in the following way:

1) Be specific

Make sure your email is brief, polite and clearly outlines why you want to work in their lab in particular. 

I would recommend searching for a few different labs that do the specific research you're interested in, which you can find out by looking for recent publications or on the websites of the relevant institutions. Perhaps someone in your current department can recommend a lab- they may have connections, or would be willing to write you a reference letter. 

2) Be persistent

If you don't get a reply within a few weeks, follow up once and then let it lie. Try asking someone else. 

Most importantly, don't give up- it took me nearly three months of emailing back and forth with ten different people when I was in a similar situation last year.

3) Be enthusiastic

Remember professors are generally happy when someone expresses an interest in their niche field and shouldn't just dismiss you offhand. 

Good luck!