Operating a pump wit a PIR sensor

Many ways to do this BUT easiest for a beginner is probably a relay.
Complexity can be added as required :-).

You can also put airpump where relay coils is on diagram and drive it directly. See below for a good FET to use.

Output voltage from PIR is not known - you will need to advise this BUT this will almost certainly work.
MOSFET (shown as a IRF511 but see below) is an electronic switch that converts the PIR voltage output to a higher power level.
This is shown as 12V but relay can be any DC voltage that suits. Usually 5V or 6V or 12V are common.

The relay contacts act as a switch - connect them across where the switch is at present.

The STP62NS04Z MODFET is available at Digikey for 43 cents in 1's in stock.
If used in the circuit below it could be used to switch the airpump directly.
It is rated at 60Aand 33V so should probably be OK. That is a FAR higer current than the pump takes when running but fun things can occur at startup. MOSFET datasheet here

For a less painful ride you should advise

• Airpump Voltage and power or current.

• PIR output voltage and whether high on detect a(as assumed here) or low on detect or ...

• Data sheet links for above would be a REALLY good idea.

Relay contact ratings have to match air-pump requirement.

If PIR output goes low on detect circuit would need to be very slightly more complex.

1N4001 diode MAY need uprating depending on pump current drain - which you are going to find out and tell us :-).

As above - pump motor COULD go where relay coils is and no relay is then needed. This does mean that PIR may be destroyed if things go wrong. Discuss.

The cheapest option would be to use a very simple transmitter like the one found here.

The problem is you'd still need to read the sensor somehow to be able to send the data through the transmitter but that could be accomplished with a microcontroller that is much cheaper than a full-fledged Arduino. Each sensor would use a wireless transmitter to talk to a base (Arduino-based) with a receiver that would then act as the web server. That might be a bit of a daunting task if you're just starting out.