Does the word proud have a bad connotation? I want to use 'proud+something' as a company and website name but I'm not sure what connotation it can have.
Well, as pride is one of the Seven Deadly Sins and was considered to be the pivotal element leading to the downfall of the protagonist in Greek tragedy, you might say it can have negative connotations.
From the Wikipedia article:
In almost every list, pride (Latin, superbia), or hubris (Greek), is considered the original and most serious of the seven deadly sins, and the source of the others.
That said, most people use it in a positive way these days. You're probably safe with whatever construction you're contemplating.
If you were to call your company "Proud Office Supplies", it would make me think of a homosexual semi-charity thing, like rainbow-coloured stationary. In general "proud" and "pride" are often associated with groups that feel under-represented in society, like homosexuals, blacks, racists, women—basically any group that traditionally likes to bring it to anyone's attention that they have favourable qualities. This connotation does not apply to any possible name with "proud" in it, but it might; that is something you should consider.