do you mean that each year she buys many presents? or, that because she buys one present each year since 2000 she has now bought many presents?

I would say either of the below with preference for the latter; My wife buys many presents for our wedding anniversaries Or My wife has bought many presents for our wedding anniversaries.

It is correct and reasonably natural. It would sound a little more natural as follows: “My wife has bought many presents FOR our wedding anniversarY since 2000.” Use “for” instead of “on” because she probably bought the presents prior to the actual anniversary date, not ON the date. Use “anniversary” singular because that is more common, at least in US English; we have only one anniversary (X month, Y day) that recurs once a year. The sentence is a little ambiguous: Do you mean many presents each year, or one present per year (adding up to many, after 18 years)?