Should I use multiple voltage regulators?
My advice here would be to try to get a 5 V power supply with at least 1.5 A output just to be sure. Then get a linear 3.3 V regulator. A heatsink may come in handy as well, but most likely won't be needed if my assumptions are correct.

Here's my reasoning: Mbed uses 3.3 V I/O, so it would be best to run digital circuits at same voltage as to not have any potential problems with voltage translation.

So run everything except the Mbed and the E.C. circuit from the linear 3.3 V regulator. 

The temp probe needs to be sampled with ADC and you'll get a bit extra resolution if you can run it from 3.3 V, since each bit of the ADC will represent smaller value. Also if the Mbed is going to sample it, then it would make process a bit easier due to reference voltage being more or less same.

The pH sensor is rated to run fine with 3.3 V operation, so just to make sure that there are no problems related to serial port high/low levels, run it from 3.3 V.

The WiFi is 3.3 V already, so no need to comment much about that. I guess that you're going to try to power it from Mbed. If that's the case, then it's good since it shouldn't interfere much with operation of other components. 

The Mbed itself has its own 3.3 V regulator, so it would go directly to 5 V power supply. 

Finally we have the E.C. circuit, which to me looks the most problematic. It's 5 V and Mbed is 3.3 V, but has 5 V tolerant I/O. If the circuit will work fine with 3.3 logic levels for input, then that's good. Otherwise some level translation may be needed. Unfortunately, the datasheet doesn't say anything. Next we have the "as close to 5 volts as possible" and "ripple and noise free" requirements which feel a bit unprofessional to me, since it isn't really explained what happens when voltage isn't exactly 5 V and there's no such thing as ripple and noise free. For that reason, I'd add a low ESR capacitor as close to its power supply pins as possible and maybe even a ferrite bead on the 5 V line, just to make sure the noise from power supply is away.