There are two complications; first is that there is no single blood draw tube, but
many. And each has a different set of additives that do things to the blood in order to make the blood is preserved in the right way for the kind of testing that needs to be done (preserving this and that, clotting, or preventing clotting, separating, etc.).
If your MC doesn't have or use the right kind of tube, that can cause trouble. Does she regularly draw blood? If she's doesn't have that history, she could screw it up. And the type of tube she needs, and the amount of refrigeration and processing it wants will depend on the drugs she's searching for. She might actually need multiple samples.
Second, if your character wants to use the blood samples for official evidence, then she'd want to get it on ice right away. If it's just for her information, that's a different matter.
As to your actual question--I found some suggestion that the type of tube with the additive used for toxicology should only sit out at room temperature for 4 hours (though that's using the blood for a different kind of test). Note that's probably the being careful so they don't screw up guidelines. But I've also found something that blood samples in a police car's trunk could sit for 7 days at or above room temperature while having only a minor affect on BAC.
So you're probably fine, but it really depends on what she wants to use it for.
Does it actually have to be in her purse? If she takes it in the hospital, I'd assume there are better ways to transport it to elsewhere in the hospital? I might even guess that someone could deliver it to the lab and put it in the refrigerator for her. If there are plot reasons that it needs to be in the purse, then okay. But it would seem really odd to me that some sort of medical professional, in a hospital, would take a vial of blood and put it in her purse.
If it does need to be in the purse for some reason, she could still refrigerate it. A hospital is going to have some icepacks--though those wouldn't stay cool the whole time, they should at least extend viability.