First, sorry to hear about this. There have been several topics on here about how Betterment makes it really hard to transfer accounts. I can say that you shouldn't have to take a distribution, or pay taxes. You can transfer IRA to IRA, but it sounds like an in-kind transfer isn't an option.
Call Vanguard back. If they can't do an in-kind transfer they can probably still complete a direct transfer. Instead of investments coming over in kind the money comes over as cash and goes into an IRA money market.
If that doesn't work you could do an indirect rollover, which is very similar to what Nalor described, but Betterment will make the check out to you and you have 60 days to get it into an IRA. You can only do this once per year. As long as you get the money into an IRA within 60 days there are no taxes due.