Is this sentence correct?

Better would be "will write" because you want the future tense here. Writes (present tense) would be okay, but is less formal.

Your first sentence has a bit of a problem. What's the intention with using "shortly after"? Better by far would be "When you see Ted".

Not a big mistake, it's understandable either way, just a little choppy to say "...the faster he writes the report." That sounds less formal, maybe a little bossy.

When you see Ted, ask him to contact us for information that will help him write his report quickly.

The whole little paragraph is in the future.

after you've seen him (in the future) ask him (in the future)...contact us (in the future), is dictated (in the future), and finally, WILL WRITE (in the future).

Change "you've seen" to "you see"

Change "he writes" to "he'll write"

Who wrote this to you ? Commas are needed before and after.

you should use writes