In the hope that you will personally review my experiences and qualifications, I have included my resume with this letter. I would also appreciate an opportunity to learn more about the position and to discuss my abilities in greater detail. Please contact me at any time to set up an interview, either in person or over the phone. I am very interested in your company and excited about the possibility of working as (Name of Position), and so I will gladly adjust my schedule to accommodate your needs. Thank you very much for your time and consideration. I look forward to hearing from you.
Side note: In my experience, most people want to write a request letter because they want to use it as an avoidance strategy. This avoidance strategy is often used to either avoid conflict or worse still, to avoid responsibility or assign blame.
When used to avoid conflict, the recipient is usually someone the writer does not get along with. So instead of taking a short walk to the table, picking up the phone or having a meeting, they just send an email.
When a project looks like it will fail, many toxic employees used the request letter to avoid responsibility or assign blame, they have the following excuse: I have sent an email to such and such person requesting for xxx. They took their time giving me xxx, so it is not my fault.
The usage of this strategy often leads to some of the worse places to work. So please dont do it. I am including this advice here because you need to know if someone is trying to do it to you. In some situation, it may be appropriate to send a letter or an email. Mostly, to outside organizations. For example, an audience with The Pope. Even then, I would try to contact someone in the Vatican before I write a formal request letter.