We bought a 2015 Toyota Corolla standard edition and the car does not have keyless entry feature. When the car is parked along with a million cars in a crowded lot, a beeping transponder would help greatly on locating the car. Can I buy something where there would be a tiny transponder sitting in the car and will come with a small remote so that when I click on the remote, the transponder will beep, letting me know the location of the car?
If your car did not come with a keyless entry system whatsoever (as in no fob or buttons) you can purchase an aftermarket keyless entry system to install in your car. If you have power locks it will be easier to install.
Shop around at dealerships and local vehicle audio shops. Often, a dealership will charge several times what a local shop will charge to install similar equipment. If you are comfortable taking apart the trim in your car, you can also install these kinds of systems yourself.
It appears that Toyota sells a keyless entry add-on system for their vehicles. I cannot find specific information about the add-on kit for your 2015 Corolla (I presume it's an L trim package). I would suggest contacting your dealership and asking them directly.
I can't seem to find a product that will simply make your car beep when you press a button. That functionality is often built into remote keyless systems. Multiple presses of the lock button will sound the vehicle horn.