They can crawl but they still provoke for moving which means standing is still probably the better option.
They can crawl while prone, but still take the -4 to AC and still provoke for movement. If the character is a rogue and has Rogue Crawl (or the feat Fast Crawl from the Pathfinder Chronicles Faction Guide), he can even take a five foot 'step' or crawl at half speed. According to the d20pfsrd entry for that feat, normally you can crawl 5 feet as a move action and cannot take a 5-foot step while crawling.
When you are crawling at 5ft/move action, you can still use Acrobatics to avoid AoO's, albeit with the DC increased by 10ft for moving at full speed, which in this case is 5ft.
Actually, there isn't much you can't do while prone. Movement is hampered, and some ranged weapons become unusable. But that's about it.
You can crawl. Note that, unlike normal movement, crawling provokes both in the square you started and in the square you finish. (And any square in between if you can somehow crawl more than 5 feet, (with Rogue Crawl, for example))
Crawling: You can crawl 5 feet as a move action. Crawling incurs attacks of opportunity from any attackers who threaten you at any point of your crawl. A crawling character is considered prone and must take a move action to stand up, provoking an attack of opportunity.
You can even use Acrobatics to avoid AoOs.
Acrobatics
In addition, you can move through a threatened square without provoking an attack of opportunity from an enemy by using Acrobatics. When moving in this way, you move at half speed. You can move at full speed by increasing the DC of the check by 10. You cannot use Acrobatics to move past foes if your speed is reduced due to carrying a medium or heavy load or wearing medium or heavy armor. If an ability allows you to move at full speed under such conditions, you can use Acrobatics to move past foes. You can use Acrobatics in this way while prone, but doing so requires a full-round action to move 5 feet, and the DC is increased by 5.
While we're on the subject, can acrobatics be used to stand up without provoking?
No. If you are prone, you must stand up (or try to crawl away), which provokes attacks of opportunity. You can attack with melee weapons, cerain ranged weapons, and cast spells as normal, but at a -4 penalty. Melee attacks made against you are at a +4, and ranged attacks against you are at a -4. (Good idea for a Cleric/Oracle upon proning is to cast Grace, which is a Swift Action, then stand up without provoking attacks.)
There are (most likely) feats or class features that would allow a character/class to do so (or to just not take attacks of opportunity from standing up), but for most classes, the value of a feat (that is, the usage and acquiring of them) isn't one to be taken lightly, especially when there are easily better choices.