Is the grammar in this article of this student constitution correct?

The grammar of the sentence is flawed and ambiguous.

... who currently or have held previously ...

It probably means "who currently or previously have held", that is:

... who currently have held such an office (i.e. have been holding such an office) for at least sixteen weeks or who have previously held such an office for at least sixteen weeks...

That could be simplified, of course, to "who have held", since the present perfect includes everything up to the present.

A "legal" ambiguity is whether the provision is enforced at a time earlier than election day or on election day. Is it a provision that determines whether you are a valid choice on election day or a provision that determines whether you can claim to be running for the office at all, weeks ahead of the election? If the former, candidacy is a status you can have (or acquire) up until voting is closed. Will you have held your office for the requisite length of time if every day up to election day is included in the tally?

P.S. We wouldn't say

... who currently or have worn previously a blue blazer

when our meaning is

... who are currently wearing or have previously worn a blue blazer

Dropping the tensed verb are wearing in the first half of that clause is ungrammatical. There is no verb in the second half of the clause that could be popped into the "gap" of such an ellipsis. However, I don't think this ungrammaticality has any real bearing on the issue. The question is, at what point do you count up the number of weeks? If you assumed a qualifying office sixteen weeks and one day before election-day for this new office for which you want to be a candidate, would you be permitted to run?

I agree with @TRomano's analysis that "currently or have held previously" is very poor phrasing, but it does not alter the fact that the natural reading of the text is that the time condition applies to both of the possibilities. The structure is clearly "Candidates who have done (x or y) for (required duration)", which means "candidates who have done X for required duration, or who have done Y for required duration."

In fact the attempt to pack the action options into the clause "currently or have held previously" reinforces this interpretation, as the only reasonable way to phrase it so that the duration applies to just the first option would be to say something along the lines of "who currently hold a position, or who have previously held one for (duration)".

In short, I think you're out of luck.