Will the crimp that happens with the bullet seating die suffice?
YES
When done "right". But doing it "right" takes a little bit of practice, and knowing what "right" is.
No, You do not need any additional dies. You may, decide you want a separate crimp die, but you do not NEED one.
What brand(s) of dies do you have, and how old are they??
I've been loading (and crimping) ammo in different ways for over 40 years. If there's a way to screw it up, I've probably done it, at least once.
Most of the important things about crimping have already been mentioned, so I'll try to hit points I don't think were, or were not mentioned, enough.
9mm Luger and .41 Magnum, these are the two classic "taper vs. roll crimp" rounds (also tapered case vs. straight wall case). Seating dies have a crimping shoulder built in, the process for using it has been described, (and will be in the die instructions too, or SHOULD be).
In revolver rounds, the primary use is to prevent bullet jump during recoil. In magnum calibers, it also has the benefit of providing a bit more resistance to bullet movement allowing for the better combustion of very slow burning powders. In semi auto pistol rounds the crimp is to help hold the bullet in place during the stresses of the feeding cycle. In ALL rounds, the primary thing holding the bullet in place is case neck tension.
The important things are "enough" crimp to do the job, and in the right place. The die shoulder "rolls" (bends/folds) the case mouth into the bullet. There must be a space on the bullet for the case mouth to go into.
Meaning the case mouth must be aligned with the cannelure (crimp groove) of the bullet. IF not, crimping is a bad thing. If there is no groove for the case mouth to crimp into, the round can be deformed. Buckled brass, to the point the round will not chamber, is possible. The other possibility is (as was mentioned) the bullet being compressed, and the brass springing back, which can actually LOOSEN the hold on the bullet.
CASES MUST BE OF UNIFORM LENGTH, or you will have to adjust the die separately for each individual round.
Take a look at a factory round to get a good idea of what the proper crimp should look like.
TAPER crimp also presses the case mouth into to bullet, but uses a gradual taper, not a sharp shoulder like the roll crimp does. It also can be overdone, actually reducing the case grip on the bullet. Taper crimping does not require a crimp groove in the bullet, because the taper does not "turn" the case mouth into the bullet. Case length is still critical for uniformity, though.
I do not use the Lee FCD die, and have no opinion about it, other than I seem to be able to get along fine without one.
I do use a separate crimp die for some pistol calibers, as a convenience more than a need. Standard advice is to only use as much crimp (either type) as you need for reliable function.
Am happy to answer specific questions, just ask!