Is it possible to accept multiple web requests at once?
It can be done. The Arduino is a single tasking environment so you have to make sure your code is non-blocking.

BlinkWithoutDelay is a good starting point (thanks @Majenko) but you should also look at the Timer1 library.

Basically your programs needs to do all the steps it can right away and whenever it does something that could block, it must do it asynchronously and during a later loop() run check if it's finished and continue with the next task. The alternative (if you know how long your task will take) is to use the timer interrupt with the library mentioned above.