Asynchronous function calls in Arduino sketch

Yes and no. You're kinda vague on what you want to do. I've made this into a few different sections (mainly focused on reading sensor data... it applies to everything but that's the context I'm using):

Threads

AFAIK all Arduinos only have one core (they can do one thing at once). For most Arduino boards, hardware multithreading isn't supported. However, there are ways to implement software multithreading. The approach by AsheeshR wouldn't work well for functions that take a long time to complete (i.e. something in a library that takes a while, or a delay) because it would get jammed up by those instructions, but it would work well for short functions like pinMode(). The Protothreads library listed there might be better at this, but I don't really know.

It'd be hard to orchestrate this with HTTP, especially since you have to make

Delays

A common way for a sketch to stall is the use of a delay. This can be solved by using, in the main loop, a if statement and the millis() function that returns a time (not a clock time, but rather the time since the Arduino started). You could also do a loop within the loop to poll sensor data.

The millis() approach wouldn't work well with things that cause the whole program to stall (i.e. more delays or loops that last a noticeable amount of time). Note that 100ms IIRC is the general maximum time to make a UI not seem laggy.

Interrupts

Interrupts are a great way to keep things almost asynchronous. They run a short piece of code (that you specify) every time a pin state changes. It breaks from the loop(), and goes back where it left off once the "ISR" has been ran. I don't have much time to explain how to do this, but a quick Google search will yield a lot of results.

As far as you example, the threading section would be the most applicable. This topic is pretty vague, so you'll have to experiment with a few things and find something that works.

I just posted some code for a task scheduler on the playground the other day, you may want to take a peek, maybe you can adapt it to your needs.

The Arduino's processor is inherently single-threaded and cannot multi-task. As has been mentioned there are however ways to create the illusion of multi-tasking. Annonomus Penguin hit on those pretty well.

Also check out TimerOne (It's probably better ;)