Middle English (in the sense 'sudden attack of illness'): from Latin accessus, from the verb accedere 'to approach'(see accede). Sense 1 is first recorded in the early 17th century
The verb access is standard and common in computing and related terminology (employees can access the office network). But its use outside computing contexts, although well established in the language, is sometimes criticized as being ‘jargon’ (we lacked adequate supply to access the markets we needed to reach). Other words or phrases such as ‘enter’ or ‘gain access to’ are suggested as ready substitutes. For another example of a controversial formation of a verb from a noun, see impact (usage).
Remember that access is spelled with a double c and a double s.