Under EU rules taxis are exempt from car seat rules, but individual countries are allowed to be stricter. In Spain, for example, they are exempt within a city, but as soon as you leave a built-up area an appropriate car seat is required. Obviously not every car can carry two of every possible seat size, so not every taxi that you walk up to at a rank can take you and your two DC to the next town.
At Palma (Majorca) airport the taxi rank has a big cupboard with a huge selection of car seats in all shapes and sizes, which are pooled among the taxis. This works well for arrivals. When you want to go back to the airport at the end of your holiday you book a taxi through your hotel 2-3 days in advance and they will turn up with the right seats. Or you can use a pre-booked transfer service both ways, but then you have to hope that the "extra instructions" that you put on the web form of "largeamericantransferbroker dot com" makes it though to "paco'sminibuses dot spain".
Also, while we all want to give our DC the best possible protection, the actual number of children for whom a seat versus no seat made a difference in an accident any given year is very small indeed. Modern cars are far safer than when this issue came to the fore, and everyone wears rear belts; the seat is often just a way to make that more comfortable. Children under 2 are probably just as safe on your lap, and if there is no booster available, just put the top bit of the bit behind their shoulder and use it as a lap belt - which is considered adequate on an aircraft landing at 200 km/h.
That said, taking your own seat is perfectly possible. Just carry it with you to the gate and any airline will put it in the hold for you at no charge, even if you are paying an under-2 fare and/or not buying any luggage allowance. I used to think this was too good to be true, but it's really how it works. If you plan to hire a car during your holiday it will save you a lot of money on renting a seat too!