We didn't get a permit for our ductless heating/air...

[ssmjlm]

I would call the company back and tell them their work won't pass inspection and you need them to fix it.

Even if you didn't get a permit for their work (which they should have pulled for you IMO), their work needs to be in line with the previous work done since that is still open and needs final inspection. (Even if the previous sentence isn't 100% accurate, I would state it to them that way.)

[emjacobs]

1) I'm firmly in the camp that reliable companies should always inquire about permits and pull them for you if you would like.
2) Regardless of having a permit, a reliable contractor should always perform work to code that will pass inspection. Yes, sometimes inspectors are extra picky and something doesn't pass, but you as a home owner know that the work won't pass inspection so the contractor should as well.

Call the company and get them to come out and fix things. For free.
Get the permit so that inspection goes smoothly.