Are there business regulations/laws/codes/best business practices that pertain to moving money from a business account to a personal account or vise versa? Is this prohibited in some way by REG E or some other regulation, etc.? Is a bank responsible for monitoring this type of activity?

How are the accounts set up?
Sole proprietor / individual
Corporation / individual?? if it's the latter, they could potentially be defrauding the stockholders.

Best practice is not to allow online transfers between accounts unless they have the same owners.

My checking to/from my savings - OK
My checking/savings to/from my sole proprietorship business account - OK
My account to/from my joint account with my wife - Still OK, in many banks
My account to/from my wife's account - No
My account to/from my single-member LLC's account - No
My account to/from my sole-owner corporation's account - No

John - regarding the single-member LLC. If I'm the sole member what the issue with funds going from the LLC to me? The transfers could be payroll which I don't see an issue with.

Because it is easier to have a rule that says no on-line transfers where the same person is not an owner of the other account. On an LLC account - the LLC is the owner. Tell them to write a check. It is much easier to police.

I can see that value of that but telling someone to write a check in 2021 may be not be well received.

I am not a lawyer and cannot provide legal advice. However, I attended a conference where a wise lawyer pointed out that the purpose of an LLC or corporation is to protect personal assets from liability. A sole-owner who constantly blurs the lines between business and personal assets similarly weakens the walls that separate personal assets from business assets. Also, the business' creditors may decide to sue the bank for allowing "conversion" of business funds to personal funds if the sole member/stockholder is dishonest and misappropriates business funds.

Bottom line, I don't care whether Randy's answer is well received or not, I'm not putting my bank at risk and you customer should be thanking you for not putting their personal assets at risk.

I am well aware that some banks and bankers would not see an issue there. And I won't say that you should not make risk-based policy decisions. But, as Randy and Brian have pointed out, there is risk to the customer and the bank when they make such transfers. And my list was given as "best practices," as requested by the OP, for banks that want to avoid gray areas to minimize risk.