Should it be "one", "ones" or "one's" in a company's slogan?

Ones in this case is a plural, not a possessive, so an apostrophe is inappropriate. Whether you need the plural or not is mostly a matter of preference/interpretation: is the company the one to call (singular), or do you call the people at the company (plural)? Strictly speaking, both are correct.

We're the ones to call
We're the one to call

Here are enough instances of We are the ones to... in Google Books to show it's not a particularly rare usage (if you follow the link, you'll see that We are the one to... is below the radar there).

I can see how some people get confused about whether it should be one or ones. Particularly Americans, many of whom are quite fixed in their belief that companies should always be referred to in the singular. Brits are more relaxed about this. But certainly there's no reason to even consider one's - unless the company in question happens to be Apostrophe Protection Society slyly trying to unload some surplus stock!

From the British perspective there's nothing at all wrong with either form because we're used to thinking of companies as either singular or plural, depending on context. In this case, the preceding We is easily enough to make us happy with the plural. In principle, those Americans who balk at Microsoft are a big company should require the singular in OP's context, but I suspect that in practice many of them would prefer to overlook the matter in this case.

Welcome, Jack. A good question, as far as I'm concerned. Yes, dodgy punctuation of we're, but The ones is fine. Consider, for example, this from 1901: She ought to blame the ones, and to punish the others.