Time- You need to schedule a portion of each day dedicated to finding and scheduling shops. This may involve checking forums like volition for feedback on companies, checking your email, and checking job boards. Of course you also have to carve out time to get to the shop location, complete the shop, and write the report.
Good grammar skills- Keep a grammar handbook handy or bookmark a website with grammar tips. Some editors have been known to return reports or reject them due to poor grammar and spelling. Don't let careless errors cost you money!
Email/Internet- I can't imagine how you can be a successful mystery shopper these days without access to email and the internet. Companies communicate primarily and, some solely through, email.
A watch- 3 words for mystery shopping: timing, timing, timing! Almost every shop will require you to record, at least, the time you are in and out of a location. Most shops require many timings in between your entrance and exit. Though cell phones with clocks/timers may work for this in some instances, I have had some shop instructions specify NOT to use a cellphone for timing because it may appear to the client that you are distracted (talking on the phone, texting, etc).
Camera- Many shops require specific photos to be taken. Again, a phone with a camera may work for some shops but others do specify that a camera with a higher resolution is needed.
Printer/Fax/Scanner- While I can see how it would be possible to complete assignments without a printer/fax/scanner, I believe having one make life much easier! You may want to print out your shop notes to take with you on an assignment and some companies to require certain documents to be printed and sent back signed. Some companies allow you to mail in receipts but most prefer, and some require, that they be scanned in and emailed back. I have not yet had to use a fax because I use the scan method but I know that some people do not have a scanner and choose to use fax instead.
Recording equipment- This is only required IF you choose to do a shop that requires recording. I am not very knowledgeable about this type of equipment as I have not done any shops that require it. I do know that some companies allow you to rent the equipment from them for the shop and return it.