You can create your own letterhead in probably 5 minutes with Microsoft word.  Not complicated.  

IF the matter involves a company, company letterhead would be a requirement for correspondence, documentation. 

use your word processing program to make something up

If you are presenting yourself as a representative of business entity, this is NOT an unusual request.  So print out a letterhead - this isn't brain surgery.

They want a company letterhead to demonstrate that you are a business entity.  You are the one asking for a job and you seem to resent the rules for getting their business.  They are not forcing you to do anything.  They have a stated policy for all to follow to apply for the work. You are under no obligation to compete.  If you are to be competitive in the marketplace you have to be better than others looking for the same work (have my doubts there) You have to have experience (I know you are lacking that).  You have to understand what the rules are and follow them.  (you don't)  Why don't you intern with a company which does what you want to do, learn a bit more about it and bid on a future contract when you have more knowledge and experience.. 

They cannot force you to create a company letterhead.

They absolutely CAN refuse to give you whatever it is you want that requires an application on company letterhead.

If they say it's a requirement, you had better believe them. 

i would hope not, maybe they can give you the money for it

you can do that yourself and save it as a template in Word

would take less time than asking on yahoo