Let's check with the travel agency.

Literal

Travel-agency [to-に] inquire-try [volitional-よう].

問い合わせてみよう combines 問い合わせる (to inquire) with ~てみる (try doing) in the volitional form (よう). The volitional ~よう expresses 'let's do X' — the speaker's invitation to act together.