I read that birth control for males is prohibited, and those who have vasectomies are not allowed to marry a Jewish woman who was Jewish from birth. However, if I am a Jewish male (from birth) already married to a Jewish convert, we already have 4 boys (all Jewish, born after an Orthodox conversion and marriage), does that somehow allow me to use specific forms of birth control? It’s important to mention that she is not able to be on birth control due to health reasons and the feeling of being done having children is mutual.
As you so correctly write, birth control for men, certainly in the form of a vasectomy is prohibited. The reason is because procreation (called the mitzvah of pru urvu) is one of the most important mitzvot, and is the first of all the commandments in the torah. The first thing G-d told Adam after creating him was to procreate and multiply. In fact, one of the reasons that a man that had a vasectomy may not marry most Jewish women is in order to discourage a person from sterilizing himself. G-d created the world because he wants it to have inhabitants. When a person sterilizes himself, in a way, he is stopping the “flow of life” to the future generation and the world.
Aside from this, you have another issue here. We are commanded to procreate and have children. This is the first mitzvah in the torah! In order to fulfill this mitzvah one has to have a boy and a girl. Therefore in your case, you still have not fulfilled. (Even after having a boy and a girl, we are still commanded to have children, as part of the mitzvah called, “shefes”, -to fill the world with people. This mitzvah however is Rabbinic in nature, and not as strict as the mitzvah of pru urvu). Therefore I would urge you to reconsider you want not to have any more children.
Regarding birth control for your wife, (not that I am recommending it, because of the “pru urvu” and “sheves” issue), after consultation with doctors, it is hard to understand why no type of birth control would be possible, as there are types that do not affect the woman’s hormones etc. and it would be advisable to speak to a doctor, and a rabbi regarding this.
Sefer Hachunuch Mitzva 559, Even Haezer 1-5,6.