Correction- she does not take birth control
Condom during sex. No breakage (you don't need to fill them with water after, they break wide open, pin holes are a myth), and then emergency contraception for her after (totally unnecessary and actually a bad idea as now she has those excess hormones to deal with and it will mess up her cycle for the next two to three months. I hope you don't go through that every time you two have sex! Maybe she should take BC pills until you get your vasectomy. But I would not worry about pregnancy here although she will possibly be late trying to process the emergency contraceptive. There is no reason to believe she'd get pregnant when the condom is used correctly and then the EC on top of it. I'd relax and not worry.