I have seasonal alergies in the spring that turn asthmatic. In a bad years I'm near hospitalization before I saw a doctor. I have not been oil pulling that long. So I have not been through the hat fever season yet on OP. When I started I had a runny nose that cleared up in a few weeks.
I my opinion as long as it isn't motor oil you can you it. The Olive-Oil you bought from the grocery store will be to start. That is was I started on. Mine was a little thick. You can shop around for other types. I like seseme, and it is one of the oils recommended to start out on. If it works out for you, then start looking around for organic, cold pressed, etc. I buy all my stuff from the grocery store, and have not had any problems.