I worked through the dependencies and thought I would post them since I didn't find a place on the internet that had the list. Again, this is on a fresh CentOS 7 installation.
[edit August 2018]
The problem still exists and the these dependencies are still needed, but I noticed that in order to install qt5-qtwebkit, you'll need to add the EPEL repo. So, before installing, add the repo:
yum -y install epel-release
yum install qt5-qtdeclarative qt5-qtwebkit qt5-qtx11extras qt5-qtgraphicaleffects qt5-qtquickcontrols
After installing the above dependencies, I was still unable to double-click the file to install it. I had to install it from the command line as root:
# rpm -ivh teamviewer_13.0.9865.x86_64.rpm
It is now installed and happily running with CentOS 7. The other cool thing the installation does is add a 'teamviewer.repo' file so updates will be done easily in the future. See the end of this page for more information.
I hope you don't have the same problems installing it, but if you do, try the above.