Is it legal for a mobile home park owner to require lot rent payment only through a sroe with a $4 processing fee with no other method of payment?
Yes, if that is in your lease with them. A lease is a contract: whatever you agree to in a contract, if it is not inherently illegal, is proper and enforceable. There is nothing illegal about requiring a processing fee (many business charge one when taking credit card payments, for example) or requiring payment in a certain form. Therefore, if it's in the lease, it's allowed.
If you have a written lease for a set term (like a one-year lease) that is still in effect (not yet expired) and the lease does not specify how to pay, you can pay any way otherwise legal (e.g. check, money order, etc.). Once the lease expires, they can change that.
If you have an oral (unwritten) lease or a written "month-to-month" lease, they can change how you must pay on a month's notice.