Without teleporting, go to bank and deposit your plague sample. Teleporting with the Plague Sample in your inventory will destroy it, forcing you to retrieve another sample from Elena.
Retrieve the plague sample from the closest bank to Rimmington and walk there.
Enter the western most house in Rimmington and talk to the Chemist "About Biohazard". Do not mention the plague sample, tell him you are a friend of Elena and that you need the paper for a guy named Guidor. (3•2•2)
After getting the touch paper, you may have to talk to the chemist again.
If you do not have the plague sample and paper when you enter the gated south-eastern area of Varrock, and acquire the three vials, you cannot simply exit the area and go back in because all three vials WILL be taken and you will have to do it all over. If you do forget the sample or paper, drop the three vials just inside the gate and run to the bank to the north. Retrieve the item and return immediately to pick up the vials before they disappear.