Your example is basically fine, although in my experience (as a native speaker) you wouldn't explicitely mention Essen as it is usually clear from the context:
Willst du noch was? / Willst du einen Nachschlag? - Nein, danke! Ich kann nichts mehr vertragen.
Do you want another serving? - No thanks! I can't eat any more.
Another option is to express it positively (i.e. without negation):
Nein, danke! Ich bin satt / voll.
No thanks! I'm full. (Be careful with voll though, out of context it denotes saturation with alcohol rather than food ...)
Other examples for vertragen would be:
Du könntest einen (neuen) Haarschnitt vertragen.
You could use a (new) haircut.
Die Suppe könnte etwas mehr Salz vertragen.
The soup needs some more salt.
Er verträgt keine Nüsse.
He cannot eat / is allergic to nuts
Der verträgt doch nichts!
He is a weakling! (colloquial, mostly used with regard to alcohol or enduring (friendly) physical "abuse")
General alternatives are:
- aushalten - endure
- (kann) (ge)brauchen - could use / need (as in the examples)
As for "much used in daily german": There are many local / dialect alternatives that of course are far more used in daily speech, but for "standard" German you should be good with the above.
EDIT: As Takkat mentioned in his comment, I left out sich vertragen [mit] (get along [with each other]). I did so on purpose, as this goes in a completely different direction. But since he mentioned it already, I wanted to at least take it up in English for everyone to read:
- Wir vertragen uns nicht so gut.
We don't get along very well.