If you just want the newer package this one time and you also want it to be replaced once the right version is released for your current release of Ubuntu you can:
Install the build tools and dependencies:
sudo apt-get install build-essential devscripts
sudo apt-get build-dep vagrant
Download the package source and extract it:
dget -u https://launchpad.net/ubuntu/+archive/primary/+files/vagrant_1.8.1+dfsg-2.dsc
cd vagrant-1.8.1
Add a changelog entry with a carefully crafted local version suffix (replace anatoly and the log message with whatever you want):
debchange --local=~anatoly 'Backport to Xenial'
This will bump the package version to 1.8.1+dfsg-2~anatoly1. The tilde suffix has the property that it is actually considered as a lower version than 1.8.1+dfsg-2:
$ dpkg --compare-versions 1.8.1+dfsg-2~anatoly1 '<' 1.8.1+dfsg-2; echo $?
0
That means the package manager will suggest to “upgrade” the local package once version 1.8.1+dfsg-2 (or newer) becomes available in the configured repositories.
Build the binary package:
debuild -b -j"`getconf _NPROCESSORS_ONLN`"
Install the built package(s) with sudo dpkg -i <DEBFILE>....
P.S.: I didn't try to build the package myself so you may run into issue. For instance, the source package for Yakkety may depend on a package (version) that isn't in Xenial (less likely at this point and since it's the same upstream version). In that case we can look for ways around this.