In most courses, the answer is yes. Some of the top links courses such as Ballybunion (Old) and Royal County Down do not allow carts on the courses due to both health & safety concerns and to protect the links.
Please see each individual golf course pages for more info. Carts are also subject to availability and courses will sometimes not let carts on the course when the weather is bad. Some courses also only allow carts to golfers with medical certs.