It's a slightly awkward phrase, because it's mixing different Englishes - 'seaside' is British English, while 'vacation' is American.
In British English, you might say 'They often go on holiday to the seaside'. Or if you want to use 'for' - indicating purpose/reason - you might say 'They often go to the seaside for a holiday' or 'They often go the seaside for their holidays'. Note that 'on holiday' is a fixed phrase - like 'at school' or 'at work' - so there is no article. If you use any other preposition, you need an article.
I'll leave it to a speaker of American English to suggest a suitable expression with 'on vacation.'