Is it jump rope or jumprope?

Put simply, 'jumprope' just isn't a word. If a person says 'they jump rope' the word 'jump' is the active verb, but it is always written as two words since it's technically a contraction of 'jump over rope'. As a verb, it literally means to "use (verb) a jump rope (noun)".

In some countries, it's flipped and they primarily say rope jumping or rope skipping which, again, is always two words. Here the phrase is a verb, akin to bungee jumping or base jumping. Linguistically speaking, ‘bungee’, ‘base’ and ‘rope’ are just the 'specifier' of the main verb.