Hey Laura,
It sounds like we are in the exact same boat. We use 365 SharePoint and Workfront. We have personally had a ton of problems with the integration and I have filed over half a dozen bugs. However, they appear to all be fixed now, so hopefully it will be smoother from here on out.
We have not actually started using an online approval process yet (still emails), but are working towards it.
Most of the employees in my organization use Chrome, which is what is so great about 365 SP. You can now open documents from any browser, not just IE. Unfortunately, you are right, from Workfront itself, you can only open the files when you are using IE, and not Chrome.
Here are some things I think could solve the issue or mitigate it slightly. I have contacted Workfront on the first three:
Fix the issue so that Chrome can actually open files from Workfront
Have a link somewhere on Workfront to take me to the SharePoint locaction of the folder I am in. Users could then open from there.
Have a link in the approvals email that goes out to users, taking them to the SharePoint location of the folder, so they can open it.
Install Chrome extensions
There is one that makes Chrome imitate IE. Unconfirmed that this works right now.
There is another one that enables you to right click on a link and then select "Open in IE".
I'd love to hear if you find a way to get around this issue also, or other ways to use Workfront and SharePoint. My support from Workfront has been somewhat limiting. At first they just kept telling me they don't have SharePoint, so they cannot test my issues. Whenever I suggest they don't have a solid approval workflow, they just suggest I use ProofHQ, which just doesn't work for us :(.