I have seen this done and know of cases when it was asked to be done. And no we do not have to carry around oil. If someone requests it then the elders can gather with the oil anointing the person and praying.
Did you read what happened? We were at the beach. We were all gathered and able to pray but annointing was requested. So that's why I say it might be a good idea to carry oil.
Actually, in the last 4 times my husband has done this, three of the times were times he had to go seek out oil. You never know when this will come up! Only one time was it actually at church - and our church is a hotel.