A camcorder is a videocámara. Assuming you can find a place that rents them, the operative phrase is "Quiero alquilar una videocámara" ("I'd like to rent a camcorder"). Then you'll have to specify what format of tape you'll need -- probably VHS (uve hache ese).
Since you're from Britain, your chances of finding a useable camcorder are probably good. NOTE TO AMERICANS: Anything you record on a European camcorder will be unplayable at home unless you have a multistandard VCR. The video standard for Europe is different than in the US and Canada.