Is it fine to assume a given orientation (...) ?
Definetly not, you can't know how your user is holding his smartphone.
(...) ask to the user to adopt a specific orientation ?
I think asking the user to change his habit is not a good thing in itself. But if in your case the portrait mode is providing a really bad experience, you could have something like :
It's better to be honest, you're not asking the user to do something because you're lazy but because there is things that are impossible in portrait orientation.
I compare with games just because games are apps as well.
If you're talking about a native app, there is no problem to use a specific orientation.
Why it is not natural to force landscape for a web app ?
When a user launch an app, chances are he's doing it from the main screen so he's not doing anything else yet. I mean he's not coming from Google (portrait) nor a blog (portrait too). He's going to use an application, which he knows, is a tool and not "just" a website. There is no antecedents and he's used to see native apps working only in landscape mode.
On the other side, when he launch his mobile browser (which is an app) he's going to navigate in portrait mode (I assume most of the people hold their phone this way). So he'll go to Facebook (portrait), then Twitter (portrait), then Gmail (portrait) then to Google (portrait), type something to find your web app and come across a blog (portrait) which has an article about your product. Finally he's here and he's asked to turn his phone, which is clearly not usual for a website. So it's in his own rights to ask "why" ? Why are you asking this to me when I can navigate everywhere else without any problem ?
Let's be honest, I think most people will turn their phone without wondering if you ask them to because they are curious and at least, they are already here. But for the others it's better to have a message explaining the why.
Btw, here is an interesting article about device orientation : http://uxdesign.smashingmagazine.com/2012/08/10/designing-device-orientation-portrait-landscape/