Sometimes just writing an email that is obviously written by/with legal counsel speeds things up immensely, in my experience.
I'm not qualified to give legal advice, but my life advice is: get legal advice if you can afford it.
The end goal is not to go to court, of course.
They also underestimate the cost of trials, which is $100k+ for anything serious due to discovery.
There are lots of situations where a letter from a lawyer gets results without you needing to do anything more beyond that. If you can afford to pay a lawyer and it won't cost more than the value of the thing you seek, I think a letter from a lawyer is a terrific idea.
My advice here would be to collect all of the documentation you can and make the clearest, most complete communication of your intention that you can through a channel that can be stored for later (email or registered mail including a dated letter with an indication that there is a copy).
Give the benefit of the doubt: "I realize that you are busy and may be trying to figure out how to proceed."
But super clear about concern, intention, and desired outcome: "I would like to exercise my option holdings at this time. I want to make sure that we are able to conduct this transaction in advance of the XYZ timeline detailed in the option plan. If, for whatever reason, it is easier for me to make this request of some other individual, group, or firm, please do let me know.
If you know the corporate law firm that represents the company, you can also reach out to them directly or indicate to management that you are going to reach out to them (not in a threatening way, but in a I'm-looking-for-a-response-and-am-not-going-away way). The law firm may actually be the option plan administrator, and they are duty-bound to administer it per contract.
Good luck!
Since you're using the words exercise and vested, maybe you have vested options in which case the policy (by law depending on jurisdiction) should tell you how long you have to buy your shares, how the price is determined and when the company has to respond with the strike price. IANAL but giving you 90 days to exercise your options and waiting 91 days to tell you how seems illegal in several ways.
1 - You should already have an option exercise form that you can fill out to actually exercise these options / notify the company you want to do so. It is important that you fill this out and have it in writing, email/certified mail/whatever, that you sent this in with a request to exercise because of 2.
2 - Depending on your employment agreement and your role, you should find out exactly what your Post-Termination Exercise Window is. This is the time period in which you have, after a certain condition, the ability to exercise your options. If you do not exercise them within this period, in most cases, the options will revert back to the company and you will not have the ability to exercise them in the future. This can often be as short as 30-90 days. While I would try to assume positive intent, you need to make sure they are not trying to wind the clock down against you.
3 - You should also be aware of the cost to exercise the options and the number of options that you have vested at the time you leave. In your grant, there will typically be several components, most importantly the strike price, the number of options and the vesting schedule (typically 4 years with a 1 year cliff). While there is much to work through with your lawyer, in general, Strike Price ($1.00) * # Options Vested (5,000/10,000) = Cost to Exercise ($5,000)
4 - Keep in mind that once exercised, if the shares turn out worthless, you can't get the funds back. The shares can end up worthless for any number of reasons including Investor Preferences, Debt Overhang, or a sale price that is too low. Overall, do your own risk/reward calculation here to make sure its worth the price of admission.
5 - I'd echo the points here in that they may never have had an employee exercise their options previously. Believe it or not, most employees do not exercise when they leave a company, and you may have been the first to do so here.
6 - Finally, document all of this thoroughly. If for whatever reason they are attempting to run out the clock on your Post-Termination Exercise Window, ensuring that you have documented you WANT to exercise them and the company didn't execute on your desire is critical in eventually getting the options.
(IANAL and you should find counsel to work through the specifics with you, this is not legal advice and there are unintended tax consequences that may occur if you don't consult a tax professional - Options are complicated!)
The tricks come later when all common shares are declared worthless during an acquisition, and no "consideration" is available to common shareholders.
I would give a couple of calls first