Skills aside, all of the characters have the potential to be equally proficient with all weapons. The difference comes in when you assign skill points; Mordecai has many skills that only affect sniper rifles and revolvers, Brick has rocket launcher skills, Roland has assault rifle and shotgun skills... and Lilith doesn't have any skills affected purely by weapon type.
What she does have, though, are skills that have effects per bullet, that have elemental effects, and that increase her firing rate. Obviously fire rate increases are less beneficial with snipers and revolvers, and 'per bullet' and elemental effects proc more often using weapons with faster fire rates-- i.e. SMGs. The intention with Lilith was clearly that she should be an SMG user, but really, you can do whatever you want if it works for you!
That being said, if you really want to play with snipers and revolvers, you might want to fight your friend for Mordecai-- he is definitely the best sniper user. Roland might also be a better choice, since his skills tend to be more generally applicable to all weapon types, and focussed on support.
Of course you can also just keep multiple weapon types with you, and switch between them as the situation requires!
In short, there are slight drawbacks to using a weapon type your character wasn't really 'made' for, but they're not so significant that they'll ruin your experience. My advice is to just play with what you find the most fun-- there's always time to re-spec, re-gear and train your proficiency with another weapon later!
EDIT: Some notes on building Lilith:
If I recall my studies of elemental effects correctly (please someone correct me if I'm wrong), the weapons that benefit the most from elemental effects are actually rocket launchers, followed by snipers. SMGs will get their effects more often in general due to their fast firerate, but snipers will get better/ more effects per bullet fired. I'd recommend a mostly Assassin build, namely Spark, Slayer, Enforcer, and High Velocity: increased bullet speed and accuracy, high crit damage, and good elemental chance.