...we decided to have a child, but only one. ...seven years later, I have a date for my vasectomy and my wife is having second thoughts.
That's fairly normal. Closing the door on a possibility - even if it was a very remote possibility - will always cause this kind of grieving.
I believe this is a decision you both should agree to, otherwise there will be lingering regret or even anger.
If she's having second thoughts I don't think you should discount them or even argue against them. Accept them, talk it out, understand what she's mourning when that door is closing. She's losing a dream. Up until now she hasn't had to face it, but now it's right there and she has to go through the grieving process for children that could yet be.
It's clear you don't want more children, but it's not clear how much she does.
Maybe she's just mourning the possibility, but was never going to act on it.
Maybe she really wants another child and she's hoping to change your mind - but it's been 7 years, so I'm guessing this is less likely.
She's your love, your partner, and what she needs now is support. You don't have to agree with her desire, but you probably shouldn't get the vasectomy if she's still having strong feelings about it.
Either way, accept her mourning, mourn with her and support her as much as you are able.
With what little I know of you and this situation, I'd approach it this way:
"I'm not interested in having more children. I think that having a vasectomy is the best course of action to avoid having you take hormonal drugs for the next decade, but I know it's hard to shut the door on the possibility of having more children. I don't want to cause any unhappiness, so I'm willing to cancel the appointment if that's what you'd like. That doesn't mean I'm any more willing to have children that I was before, but it's a decision we both have to agree to and I respect your desire for another child even if I'm not willing to act on it."
"Should I cancel the appointment?"
If all she needs is recognition that this is hard for her, she will probably have you go through with it, but she will hopefully feel that her feelings are being considered and understood.
If she really does want another child, she'll say cancel, and this opens the door for further discussion. You two should really hash this out.