What is the term used for people who get drunk easily?

Not single words (although I've seen the second one hyphenated), but “cheap drunks” and “two pot screamers” are people who are/become “easily in their cups"/easily drunk.

cheap drunk (from Urban Dictionary):

Someone who easily gets drunk. They could have 3 shots of a 40% strength alcohol and they're gone. The person is cheap because they don't have to spend as much money on alcohol to get drunk.

two pot screamer ‎(plural two pot screamers):

1. (Australia, colloquial) One who becomes talkative or rowdy after consuming a relatively small amount of alcohol; one who quickly becomes intoxicated. (Wiktionary)

"Can't hold their liquor" or "Can't handle their ale" are in common use in the UK.

Can't hold their drink/liquor

if someone can't hold their drink, they get drunk after drinking very little alcohol

www.idioms.thefreedictionary.com