Post SOAP, the NRMP will publish an open slots list. The list is fixed, they will no longer update it. The list will contain contact information for each program.
Usually, applicants just contact programs directly -- either by phone, email, or fax. As you can imagine, it's a bit of a nightmare for programs, as they could get 100's of calls. So you'll find that many phones get disconnected, or a message is left that the spots are filled / will not be filled / etc. So, expect a very low pick up rate.
If a program is interested, they will want your info. You can either send it manually (i.e email/fax), or you can apply via ERAS. ERAS closes on May 31, so all of your information and documents remain there and can be sent to any program until then. Standard ERAS rates apply. I wouldn't just send applications via ERAS to programs with spots listed -- the chances that they go back into ERAS to look, without knowing your app is there, is low. Although perhaps some programs will instruct you to do just that (which is fine, if that's their process).
If you're sending your app manually, you may not have your LOR's. Your choices are to contact your LOR writers and have them send copies, or use ERAS to send an application to the program.