Is 'take a sauna' the correct expression?
Wiktionary has an appendix that includes collocations of 'take' and 'have'. It includes "'take'/'have' a bath" and "'take'/'have' a shower". 'Take' is more common in the US, 'have' in the UK.
The good news is that both terms are intelligible on both sides of the Pond and that most civilized people on either side wouldn't hold it against you that you used the less common, less preferred version.