It seems insane that this isn’t default behaviour but in SharePoint on Office 365, if you click on a PDF in a document library you’ll get the option to save the file but no option to open it.
In an on-premise implementation you just set File Handling to permissive in central admin: 
The sub-client I was talking to yesterday is one of many we at Resonate IT have helped recently who are on Office 365. Unfortunately in SharePoint on Office 355 you don’t have that option and you can’t (yet) use PowerShell. So what to do? Well after a lot of people (myself too) created a stir, Microsoft rolled this into a feature.
If you go to to site collection feature you’ll find this option:
Switch it on and you’ll get a lot less hassle from your users!!!
P.S. This works for PDF but not for HTML 
Andy Noon - SharePoint since 2001 
This did not work for my client.
Sorry to hear that Jeff. There’s always one client with extra ‘issues’ isn’t there 😉
In my case, this feature is not enabled but I can open the PDFs in browser but not one of my users in the same site collection. Any ideas?