The verb "bade" (past tense of bid) is used in older or more formal English to mean "invited" or "greeted". It is followed directly by the bare infinitive (the base form of a verb without “to”), or in this case, an adjective acting as a complement.

"Welcome" in this sentence functions as an adjective, not a verb. It describes the state in which they were received. Saying "He bade them welcome" is idiomatic and grammatically correct.