Grovel comes from Middle English: back-formation from the obsolete adverb grovelling, from obsolete groof, grufe ‘the face or front’ (in the phrase on grufe, from Old Norse á grúfu ‘face downwards’) + the suffix -ling .  So, basically, grovel comes from the Vikings and is probably how they liked to see the people they were conquering, lying prone, face downward.  You grovel when you want something, but also when you're afraid.  Definitions of grovel. verb. show submission or fear. 

OED's earliest evidence for grovel is from 1605, in the writing of Joseph Hall, bishop of Norwich, religious writer, and satirist.  However, if it comes from the Middle English, that period is from 1150-1500.  

Examples of grovel used in a sentence: 
"If they want me to get down and grovel on the floor, no." 
"I refuse to grovel to him" 
"I had to grovel to the manager for a loan" 
"She saw him grovel on the floor, begging for mercy."  
"If they want me to get down and grovel on the floor, no." 
"I refuse to grovel to him" 
"I had to grovel to the manager for a loan" 
"She saw him grovel on the floor, begging for mercy." 

While I hate to question OED, my searching turned up grovel being used by Shakespeare in Henry IV Part 2, which was written prior to 1599. The reference came with an acknowledgement that Shakespeare used grovel by back forming from groveling, which as @Bill W has indicated, already existed in various spellings.