I'll be applying in Late June and started formally asking for them around March 25th. I just received my first of four via Interfolio this week.
Many professors are incredibly busy, especially towards the end of the semester and finals week, so it's best to get the requests out as early as possible. The latest I would ever ask for a LoR would be 3 weeks prior to deadline, but preferably asking for them ~7-8 weeks prior is always my goal.
When I asked for them, I emailed asking if they'd be comfortable writing one and saying why I believe their letter would be a strong testament of my capabilities, and ended with a request to meet in person at their office if they were willing to write a LoR. If I knew their office hours already or had a closer relationship, I skipped the email part and just asked in person. During the meeting, I took a pre-stamped and pre-labeled manilla envelope (Interfolio's address), and a packet including the Interfolio LoR request form, my unofficial transcript, my CV/resume, and a supplemental sheet with a short blurb of why I want to be an MD, the due dates, and generic instructions. I also took some old assignments I had done for the Prof if I felt it would help them. We talked about my career goals, from what standpoint they could write the letter, and I asked if there was anything else they'd like to know about me before writing the letter.
They all remarked how delighted they were to receive everything in a clean, organized manner and how I approached 2-3 months in advance. The way in which you present your request is going to be the most recent memory/impression writers have of you, so make it a good one.