Yes anyone working for the HOA has the right and permission to go onto your property to look for violations.
The Home Owners Association staff can also go onto your property and mow your grass for you and then fine you and put a lien on your home until you pay up the fines and the cost to mow the grass
When you buy a home in an HOA neighborhood you sign paperwork that gives the HOA permission to go onto your property.
You're agreeing to the HOA rules and regulations when you sign the paperwork to buy a property that has an HOA.
I would never buy a home or property in an area that has an HOA because they are hell to deal with and you'll never win.
Everything in an HOA area has to be kept up to the HOA standards and if not then you'll be fined and they can then sell your home and force you out if you do not pay those fines.
Also if you continue to violate the HOA rules you can also be forced out and the HOA can take your property.