Vectorize Weathering in Illustrator

I've designed well over 30,000 tshirt designs since 1989 in illustrator, and I have to say that all the above answers are not "wrong", but also may not be correct depending on how you work and how you do separations.  There are plenty of autotraced distress / grunge files out there for free, and you can even do your own with a bit of work between photoshop and illustrator (find a texture/photo, open in psd, pull the contrast up to the points of a near posterize level, save and autotrace in illustrator). I generally mix more than distress vector to make a more realistic cracking effect.

The easiest: find some of the grunge and/or distress ai files on the web, color them white and place them over your design. This works if you want the entire design distressed. If you are just wanting parts of it distressed you'll have to position the distress into the correct above/below order (or layers, even though I do complicated illustrations without every touching making more layers). Sometimes you'll need to make the distress texture a compound path and subtract it from the portion of the artwork you are modifying, which can get tricky if the texture has too many small points. I still do separations from illustrator/distiller, but I am in the minority... its a lost skill. I still do 3-8 designs a day, all in a super hurry (approx 15-20 minutes each), so I usually keep it simple and just paste a distress over the design.

The less easy: you can do your work in illustrator and move it to psd and go insane with layers of texture effecting the artwork in a variety of ways. This means you HAVE to use something like accurip for seps. For example you could draw a basketball with wings and place the texture of an eagle over it in psd, screen it thru layers, and make seps. All depends on your time frame. If you have hours per design you can do almost anything. If I had a DTG I would.

Lastly, if you have clients wanting mockups on real shirt images, you'll need to eye drop the shirt color into the texture, but it rarely works well since the shirt color is not consistent. I give my clients mockups on vector shirt mockups.

you can add textures to your vector art and embed them. i do this. you use the opacity panel.

here watch this:

Creating and Using Textures for Design

Try a site such as https://www.vecteezy.com/free-vector/grunge-brush-strokes

You could also use default artistic brushes in Illustrator such as Chalk Charcoal, Ink, or Paint Brush, followed by using the effects gallery and applying something like a sprayed stroke.

You could  also use gradients and blend modes like Sew Heidi  shows here:

Distressed & Faded Denim Effect in Illustrator - YouTube

You could try and search for "Grunge"

Which version of Illustrator are you using?

Also, may you please show some examples that illustrate what you are looking for?