Rambam lists the prohibition to castrate a man or animal in the negative mitzvah #361. The previous negative mitzvah (#360) Rambam lists that it is forbidden for a castrated man to marry a Jewish woman.
So, can a Jewish man who had a vasectomy marry a Jewish woman?
Rambam in Hilchot Issurei Biah 16:1 rules that one who falls into the category of petzua daka or kerut shofcha may not marry a Jewish-born woman, although he may marry a convert.
In 16:3 he clarifies that these categories include injuries to the vas deferens, and in 16:7 he states explicitly:
נִכְרְתוּ חוּטֵי בֵּיצִים אוֹ אַחַת אוֹ שֶׁנִּדָּךְ אוֹ נִפְצַע הֲרֵי זֶה פָּסוּל:
If both or one of the vasa deferentia are severed, crushed, or wounded, he is unacceptable.
(In 16:9 he further clarifies that these rules only apply to one who became this way through injury, rather than being born in this state.)
So, to summarize, yes, a man who has undergone a vasectomy may not marry a Jewish-born woman, but he is permitted to marry a convert.
One caveat that is worth mentioning: if the vasectomy is carried out on the portion of the vas deferentia which is inside the body cavity, rather than within the sexual organ, Chazon Ish Even HaEzer 12:7 was of the opinion that this would not render him a petzua daka, and he would still be permitted to marry a Jewish-born woman.
