For job search purposes, can I ask my current employer for a letter of recommendation or to be a reference?

As a rule, rather ask co-workers and the HR department to act as referrals. You can try to get a letter of recommendation from your manager at salary or performance review times for your file. You may also state that you would like to become a member of an organization (make sure that you do) and needs a letter to state that you are trustworthy, on time etc.