Both my spouse and I agree on this, so we can’t possibly be wrong, 
but since I’m insecure…
What’s right:
1)“I would take action when there become specific reasons to do so”
2)“I would take action when there becomes specific reasons to do so”
3)“I would take action when there becomes specific reason to do so”
?
None of the three!
‘I will take action when there becomes specific reason to do so’ or ‘I will take action when there become specific reasons to do so’, but the former is prefered as it’s the generic singular.
or
‘I would take action when there became specific reason to do so’ - subjunctive.
heh. Actually, in the original it’s more like "people can take action when there become(s) specific reason(s).
sloppy editing on my part!
Well, in that case, what is with the “become”?
Make it "“People can take action when there is specific reason to do so,” (or “when there are specific reasons to do so”), or if you really need to stress the future orientation, use something like “when specific reason to do so arises.”
Or, to return to your original versions, only 2 is clearly ungrammatical. I can’t say I like 1 or 3 very much, however.
There’s usually more than one correct way, but #3 strikes me as the most correct of the three.
Wouldn’t 3 have to say “becomes a specific reason”?
I’m going to assume you want an answer that’s actually about English grammar, not just amateur style advice.
As far as number agreement is concerned, both (1) and (3) are correct: the former has a plural noun agreeing with a plural verb form and the latter has a singular noun agreeing with a singular verb form. (2), on the other hand, has a mismatch between a plural noun and a singular verb. This is entirely analogous to the following set of simpler sentences: (1’) and (3’) are both standard English. (2’) is acceptable only in casual/colloquial English.
(1’) There are beers in the fridge.
(2’) There is beers in the fridge.
(3’) There is beer in the fridge.
However, there is a factor quite independent of number agreement that makes all of your sentences sound odd. In particular, become is a transitive verb, which sounds odd in this construction, especially when there is no object. Compare:
There arose a problem. --> A problem arose.
There stood the man. --> The man stood.
There appeared the message. --> The message appeared.
There exist several reasons for doing so. --> Several reasons for doing so exist.
There became specific reasons to do so. --> ??Specific reasons to do so became.
The problem is that with become, you need an adjectival or nominal complement:
The reasons to do so became apparent.
The reasons to do so became my only hope.
So we could try salvaging the sentence like this:
(4) I would take action when there become apparent specific reasons to do so.
But now it’s getting pretty awkward, so I would just rephrase, and that is my (semi-)amateur style advice.
So in executive summary: As far as agreement goes, only (1) and (3) are right. As far as verb argument structure goes, none are right.
I’m going to agree with the general thrust of the responses so far, and say the best way to state this sentence would be:
I would take action when specific reasons to do so arise. or
I would take action when specific reasons to do so appear.