I know Kohanim have certain restrictions on whom they may marry. My question is, are Kohanim restricted onto a Shifcha?
הָעֲבָדִים שֶׁהִטְבִּילוּ אוֹתָם לְשֵׁם עַבְדוּת וְקִבְּלוּ עֲלֵיהֶם מִצְוֹת שֶׁהָעֲבָדִים חַיָּבִים בָּהֶם יָצְאוּ מִכְּלַל הָעַכּוּ''ם וְלִכְלַל יִשְׂרָאֵל לֹא בָּאוּ. לְפִיכָךְ הַשִּׁפְחָה אֲסוּרָה לְבֶן חוֹרִין.
Servants that have been immersed for the sake of servitude and accepted the mitzvot in which servants are obligated, have departed from the category of gentiles, but have yet to enter the category of Jews. For this reason, a maidservant is forbidden to a free Jew.
So not only kohanim, but any free Jew (i.e. one who is not an eved ivri) is forbidden from marrying a shifcha kena'anit.
As is clear from 18:1-3, a kohen would additionally transgress the prohibition against him marrying a zonah; a shifcha retains this status even after being freed, at which point she would be permitted to a non-kohen.
A convert or a freed Canaanite maidservant are Zonot in all cases because they didn't start out Jewish. The source for this is found https://judaism.stackexchange.com/a/22496/22145