First off you will definitely not walk out of the airport and find the average taxi has child seats. In the past we have used...
http://www.lastminute-transfer.com/
for transfer to Playa Blanca and found them excellent, recommended. However, note that we did not require child seats, although to the top left of that page you will see a reference to child seats. Hope this helps.

we also use last minute, we prebook online and request baby seats,then pay on arrival at hotel.
have used them the last 2 years travelling with baby and toddler

Hire a car from the airport, it will be cheaper than 2 taxis and they will provide child seats. Or take your own.

The taxis waiting outside arrivals will not have child seats in them. Either book a transfer via someone like LastMinuteTransfers or Canarias Puerto Mar who will provide you with a child seat, if you prebook one.

As robert says, booking a hire car may work out cheaper if only going for a week as a taxi is going to cost €45-50 each way. If you do, don't use Goldcar or Orlando! Try Autoreisen, Cabrera Medina or Cicar.

I'm going in March, I've booked a car with autoreisen for 78e, it will be about 50e each way in a taxi.

Many thanks for your helpful advice!