Hello!
The reality is, most people who write recommendation letters/serve as references understand that this is part of their job. Therefore, they are more than happy to work with you so long as you provide them enough time. The best way to handle this is to reach out to them asking if they wouldn't mind writing you a recommendation because you believe they would be able to speak to (x) qualities that you have. In the case that you have to submit the letter yourself (rather than having them send it), when you ask them (NOT afterwards) tell them that you give your word that you will not read the letter should they not want you to, that you will just pass it on to the necessary recipient. If they believe in you enough to write your recommendation, they will trust your word. Most professionals understand that different programs have different methods of receiving materials and will understand that certain programs require all materials to come from the applicant.