Learn Spanish (Latin America): Esperamos que haya respeto en la oficina.

You're right, hay normally means "there is" or "there are". However, because we are expressing a hope using esperamos que (we hope that), we have to use the subjunctive mood for the second part of the sentence. Haya is the subjunctive form of hay.

Do I always have to use que after esperamos? In English we just say "We hope there is..."

Yes, in Spanish you must include que (that) to connect the two ideas: "we hope" (esperamos) and "there is respect" (haya respeto). While English often drops the word "that" in these situations, Spanish never drops que.

Does esperamos mean "we wait" or "we hope"?

The verb esperar can mean "to wait", "to hope", or "to expect". When it is followed by que and a subjunctive verb (like haya), it expresses a desire or expectation for something to happen, so it translates to "to hope" or "to expect".

Could respeto be a verb here, like "I respect"?

It looks exactly like the "I" form of the verb respetar (I respect), but here respeto is a masculine noun meaning "respect". You can tell because it comes right after haya (there is), which is used to declare that a noun exists.