WASHINGTON—Donald Trump for the last week has been asking for support from African-American voters who have long backed Democrats, but his campaign has for months rebuffed invitations from supporters for the Republican presidential nominee to appear before black audiences.

The result is Mr. Trump hasn't campaigned in the African-American communities he now says he wishes to win over. Supporters of the New York businessman say they have asked him to speak at black colleges, in black churches and to the National Association...