Can I get a weather briefing and file a flight plan with ForeFlight on the web?

Yes.  To do this, tap Maps > FPL > enter your flight information.  Tap "Send to Flights", you will be taken to the Flights page and your flight information will be loaded.  Confirm all information, then tap the "Briefing" button to receive a briefing.  Tap the "Proceed to File" button to move to the filing page, confirm all information, enter additional information (for example, Remarks to ATC), then tap the "File" button to file your flight plan.

Briefings and flight plans filed on ForeFlight on the web will be synced automatically with the devices signed into your ForeFlight account.