Would you really send a text message in formal language? ASAP is commonly used in casual business communication. If you were speaking (formally or semi-formally) you'd say the whole phrase 'as soon as possible' rather than saying ASAP. If you were speaking to a coworker that you're comfortable with you might just read the acronym. It gets a little more difficult as you move to shorter communications. For an email, if you're talking to someone that you don't have a more casual relationship with, it would be better to write out the phrase. Text messages, twitter, chat programs get a little tricky. Usually these communication methods are considered to be more informal because the goal is to get information out quickly rather than worry about polite necessities. In general, I'd take your cue from your client. If your client is using formal language, pretend you're still communicating via email and mirror his/her language.

It is informal. It would be a little safer to spell it out, "as soon as possible." Keep in mind that you can't really make it polite. You can use any words. You can soften it by saying "please." It is still a command, an order. You are applying pressure. Do it as soon as possible, even if it is difficult or inconvenient for you. "At your earliest convenience" is more polite, because it is asking for less. At your earliest convenience" is not applying any pressure. Of course, if you need it as soon as possible, you'd better say "as soon as possible!"

I can't speak for customs elsewhere, but in the US, it would be perfectly appropriate to use "ASAP" in a business-related text or email. But say "please" along with it. For example: "Please get the report to me ASAP." Or, "Please respond ASAP." You can also write out "as soon as possible" in a business-related email or text to be a bit more formal. Same rule applies, use please along with it to be polite.