Are there any prohibitions against or stipulations for conducting a marriage between a woman and the man who recently impregnated her, assuming they are otherwise fit for marriage.
Shemot 22:15 states:
וְכִֽי־יְפַתֶּ֣ה אִ֗ישׁ בְּתוּלָ֛ה אֲשֶׁ֥ר לֹֽא־אֹרָ֖שָׂה וְשָׁכַ֣ב עִמָּ֑הּ מָהֹ֛ר יִמְהָרֶ֥נָּה לּ֖וֹ לְאִשָּֽׁה
If a man seduces a virgin who is not betrothed and lies with her, he shall provide her with a marriage contract as a wife.
If pregnancy is assumed as a possibility of sex, this should mean this is not only permissible but mandatory.
Short answer -- it's allowed.
You'll find a discussion elsewhere on this site about if someone impregnated a woman, do we recommend that he marry her. Which implies that the rabbi can certainly do such a wedding.
The discussion becomes a little more complicated if the pregnant fiancee was not Jewish at the time, and the rabbi now wants to convert her and marry them. (See Igros Moshe EH4:47).
The only question remaining is exactly what word to use in the kesubah (marriage document); theoretically you could refer to the bride as a "maiden" if she was never before married, and all her past partners were Jewish; there's a recording on yutorah.org from Rabbi J. David Bleich who said it looks wrong for a rabbi to sign his name on that language if the bride is visibly pregnant, so instead the text should just refer to her as an it'sa ("woman"). But if the whole conversation revolves around exactly what words to put in the document, it's clear that doing such a marriage is allowed.