Mr. X stated that Company A currently has really strong relationships with Phrma and the European Union, but have not yet worked deeply with AmCham.

have: This is correct in British English. In American English, companies are treated as singular subjects, and it would be "has." Your answer depends on your audience, but it should match the first "has" (or have) earlier in the sentence.

deeply: This is grammatically correct, but it's an odd co-location; you don't "work deeply" with another partner. English speakers would know what you mean, but we would more naturally say "closely" or "extensively."