Whether you're using the US term 'vacation' and or the GB term 'holiday', both words function in the same way. 'On vacation' and 'on holiday' are set phrases, like 'at home' or 'at work', and there is no article. If you use any other preposition, you need an article or other determiner (a,the, my,this etc) - 'for a vacation', 'without a vacation', and so on.

We also say "going on a vacation" here in the US

'I usually go to the seaside for a holiday'. 'I usually go to the seaside for vacation.' Sound perfectly fine to me. Another way to say it is: 'Take a vacation', 'Taking vacation' Examples: 'Every year I take a vacation to Hawaii.' or 'This year I'm taking a vacation to France.'