Can I say "I emailed you (with) a brief introduction of our program along with the application link"?
Both options are acceptable. When we use the preposition "with" in the context of emailing someone, it is meant in the sense of "having or including". So if you say
I emailed you a picture of my dog
then you are indicating that the picture of your dog was the entire reason for your email. The email was the means by which you sent the picture of your dog.
If you say
I emailed you with a picture of my dog
then the picture of your dog might be a secondary concern. Perhaps you were emailing your friend with a letter of correspondence, and the picture was an added bonus.
So, if we take your original message, if the main point of the email was to send "a brief introduction of our program", then it would be better to have that as the direct object of "email", (without "with") especially since you are sending it "along with the application link", which sounds like the extra feature of the email.