If you are living in rented accommodation and wish to engage a house sitter to mind your property and animals in your absence there should be no reason why you can't do so. Check your written lease agreement for any sub-clauses that may prevent you from having a house sitter in the property. Usually, anyone is allowed to stay in a rental property with the lease holder's permission. However, if you want to charge your house sitter a fee to cover rental payments you are effectively subletting the property and are on much murkier legal ground.
If your property is not covered by a rental agreement then it would be courteous to seek your landlord's permission before leaving your home in the care of a house sitter. It is widely agreed that an occupied property is far more secure than an empty one, so your landlord should be happy to consent to the arrangement.