Accessing barometer information
Sadly as of 2015 it's not possible to access barometer information in a native browser environment. A lot of work has been done on writing sensor API specifications, but as barometers are quite a rare hardware feature still they have not been receiving a lot of attention.
It is however possible to bundle your web application in Cordova and use the barometer plugin for android to access the barometer from your code. This however means that the user has to install your application and this would only work Android.
Using GPS information after all
For the large majority of applications where you need altitude information there is only one possible altitude per coordinate pair (this is for example not true if you wish to determine the level you're at in a building, but current GPS altitude information isn't precise enough for that normally either way). In those cases you can request the GPS location and simply request the altitude from a service with a surface database. Two such services are the Google Elevation API (entire world) or the NED Point Query Service (US only?).