Hi Karen,
This question really brings up several issues. 1. The only way that the agent could accept the offer is if he actually owned the property and there should have been some indication of that on the sign or in the written ad showing agent interest.
2. If this was the listing agent that you contacted and he agreed to pass the offer on to the seller, you may have felt that he was acting on your behalf (an implied agency relationship). I can understand your confusion.
In the State of Illinois you can act on behalf of both the buyer and seller but you need to be VERY CAREFUL on how you handle it. Section 15-45 of the Illinois real Estate License Law requires that the Real Estate Agent can act as a dual agent only with the informed written consent of all clients. There is a special format that you have to follow that outlines what a licensee can do for clients when acting as a dual agent and also what a licensee cannot disclose. Since the Realtor is acting on behalf of both the buyer and seller you would get less than full representation. If you feel uncomfortble with this dual representation you should not sign this document and the licensee should refer you to another licensee to help you move forward with your offer.
3. Putting the dual agency issue to the side, the realtor representing the seller should take the offer to the seller who would then make the decision to accept one offer or the other or to go to a multiple offer situation where all parties are informed that there more than one offer on the property and to redo their offer with that in mind. All decisions are made by the seller.

I'm confused by your statement about "making an offer to the Realtor" and you said that he accepted it? Was this the Realtor's property? Accepted means he signed it.

I'm thinking that what really happened was that you made an offer and the Realtor took your offer and presented it to a seller. If that's what happened then "yes" all offers must be presented until closing. Now was the Realtor a dual agent? Meaning, did this Realtor have the listing on the property? If so then that Realtor should have disclosed to you that he was working as a dual agent. He represented the seller and he also represented you, the buyer. Dual agency is legal in Illinois, but it "must" be disclosed.