Can you rely on local tourist information centres to find accommodation in the summer in the Czech republic?

The answer depends on the city. In Prague there are 1000+ hotels, not to mentions the dozens of properties on AirBnb. While it's common to see 90+% booking rates on Booking.com, there will always be some hotel with free rooms at a reasonable price. No need to go to tourist information centers, just book online.

In a place like Cesky Krumlov the selection of hotels is very limited. The local tourist information center might help, as there are many hotels in the country-side without proper online booking. However even in the worst case scenario you can always go to a nearby city - in the case of Cesky Krumlov I've simply booked a nice hotel at nearby Ceske Budejovice.

Therefore my advice is not to bother with pre-booking (unless you really care about staying in city centers) and simply finding the closest free hotel on the spot.

While travelling by car and not knowing exactly which day you will arrive in certain places, it will be helpful to use Tourist Information Center services in each location. No charge and they will know what is available in town and very likely will even make a phone call to check if they have a room available. Tourist Information is usually located on the main square of the town. Because you have a car, it is not important for you to stay right in the Prague. Find on Internet map of the Prague parking areas at the end of the Metro and continue by Metro to the centre. Using car in Prague is not advisable for tourists. Police are very fast with towing tourists parked cars away anywhere you would like to go/see. If you do want to be accommodated in Prague, book your stay using booking side on the Internet where you can also read reviews AND make sure that they do have a parking available.