I've never worked anywhere that would let you purchase before vesting. You accept a new position that came with 20,000 options vesting over 4 years with a one year cliff. At the end of one year, 25% of your shares (5,000) are now vested and you may purchase them. The company might have a valuation for them that's higher than your purchase price, but they are still typically worth "nothing" in that you can't sell them (assuming you're at a startup that's not yet publicly traded.)

Now the remaining 75% of those options might vest monthly over the next three years. "Vesting" does not impart value. It's not indication of whether you're going to make any money at all. Vesting is an instrument used to make sure you stay on with the company for an appropriate amount of time before you're allowed to own part of the company at that discounted rate.

Point: no, you cannot exercise options at grant time, you can only exercise once vested.