Some of the teams that travel a lot have arrangements with specific airlines: in the UK that’s often Titan, sometimes British Airways CityFlyer, and KlasJet is also used regularly. Some airlines have an ad-hoc charters department that will arrange things, while others work through air charter brokers. UK charterers usually have to try to source UK-registered aircraft first (and only go overseas if nothing suitable is available), and I believe a similar parallel is true for EU-originating charters.
In terms of using a specific airport, yes there are the obvious issues like runway lengths/widths, taxiway and parking stand restrictions especially for aircraft types that don’t regularly use an airport (by the way its not just length and width, its things like pavement strength too). Another issue to be resolved includes arranging handling agents at both ends (they supply steps, ground power, potable water, load and unload baggage, etc). If an airline doesn’t use an airport regularly, does it have a contract with any of the local fuel companies (taking cash in a brief case isn’t unheard of)? What about customs and immigration - are they available at the planned arrival and departure times (if not flights sometimes have to go elsewhere)? Indeed will the airport even be open at the appropriate times - after an evening match is sometimes problematical if an airport closes overnight. If the aircraft is higher-capacity than others that use the airport, will the fire and rescue coverage be adequate - or can it be stepped up on prior arrangement? The UK AIP (Aeronatical Information Publication) pages for many airports detail restrictions on specific - usually large / high-capacity - aircraft types.
There’s more but I suspect you get the picture. A good charter broker will sort it all out for the business wanting to charter an aircraft, as will a good airline that is used to doing ad-hoc work.