I created an application that can looks into the database for people having their birthday today and then send mail to them. The issue is that I want the system to do this automatically instead of me clicking it every day. I tried a scheduler but it stops working after some days. Is it possible for me to use a service to run this application.

Scheduler set to work once every day should work - what errors do you get from the scheduler?

The scheduler was set to work every 5 minutes, the only errors i get is that the 'scheduler failed to run'. I have deployed this for over 3 clients and it keeps stopping after running for like 3 days.

Is it possible to achieve this using services?

I am sure it is possible, but keeping it simple is almost always the better answer.
Why do you have the scheduler running every five minutes? Either today is my birthday or it isn't, it won't change 5 minutes from now...

Well, the emailing facility is more than just the birthday. I also want to send emails to superior officer when a transaction is done and requesting approval.

Personally, I would figure out why the scheduler is failing.
Google: Programming Windows Services VB.Net for more info and tutorials

A point of interest. In my VS2010 it says I can create a new project as a windows service. Perhaps you can recreate your app directly as a service, and put your own timing and scheduling in it?