Great question, Duke, and I'm surprised nobody has answered you yet. You first need to understand who your decision maker is for your particular service. Some decisions are made by the property manager, some by the maintenance supervisor, some by the regional manager, and some even higher up the chain. A lot of that is going to also depend on whether this is a service they already have, and if it is a service they have a portfolio based contract for already. For example, a company might contract one landscaping company to handle all their properties in a given area, so the property manager would have no control over that decision, and even taking away one property would be difficult.
The next part is how to approach them. I would say before you do that, you potentially need to lay a little ground work. Volunteering and becoming active in the local apartment association can help people start to recognize you, and smooth the process of getting a meeting. Just be careful not to make the #1 mistake most new people make, which is immediately start trying to sell at a networking event, rather than simply getting to know people.
You can also do this process really effectively online. If your association has a fan page or group, start participating in that, just so people start recognizing your name.
As for how to make that actual initial contact, I have heard that regionals and up prefer to be contacted by email. Simply showing up at their office will probably not be a good move. As for on-site, I really don't know, so hopefully someone else will chime in, but I do know that goodies are always a good idea, assuming their company allows them to accept them. You might even try the soft approach, where you walk in not with the intent of talking right there, but simply to set up a meeting. Bring some cookies with you, and I think you would have pretty good odds with scoring a meeting a bit later. Bringing home baked cookies might be good since you are just starting up, and they also send the message that you took some time to make them.
Hopefully there are a few ideas there to work with, and maybe someone else will chime in with some more!