Looking among the fleets in Westeros,
- Highgarden does not have a navy, however the Arbor Navy, one of the largest naval force in Westeros answers to Lord of Highgarden through Lord of Arbor.
- The North does not have any navy, however the White Harbor Navy is under the North command through Lord Manderly.
- It is not specified if Riverlands has a navy other than its river capable boats, however, Mallisters have a navy to protect themselves from Iron Islands.
- Dorne has a formidable naval force, however it is not specified if it answers directly to the Prince of Dorne or through vassals under Dorne.
- To the best of my knowledge, Westerlands lost their navy during the Greyjoy Rebellion. Whether vassals under Westerlands have a navy is not specified.
- Gulltown had a small navy during Robert's Rebellion, however it is unknown if it still has one. Also, it is unknown if vassals under Vale or the Vale itself have any navy.
- The navies of Stormlands or its vassals are not specified.
Finally, the most important one:
- Lords in Iron Islands have their own navies, some of which are more formidable than navies of the Great Houses. Moreover, the Iron Fleet is directly commanded by the Seastone Chair, i.e., Pyke.
In conclusion, some of the great houses have navies through their vassals and some of them have their own navies. Crownlands in principle is similar to other great houses so it may or it may not have its own navy. One more point is that Crownlands, just like any other great house, can summon its own army. In the non-canon RPG game, their military capability is specified as 15k, however considering the population of the King's Landing, it can be much more.
Real World Examples
Most of the great empires had drawn most of their land forces from local lords, however, many of them had their own standing navy in medieval ages. I know English Navy and Ottoman Navy are two examples proving that point. The only example I know where vassals provide naval force is Vikings.
I believe it is due to two reasons.
- Ships are very expensive and building navies require careful planning for extended periods, which may not be available to local lords.
- Naval personnel is much harder to train, i.e., you can expect a peasant to form a three deep spear formation, but you cannot expect a peasant to operate on a ship. For example, after the disastrous Battle of Lepanto, Ottomans rebuilt their navy in six months. However, since they did not have experienced seamen they did not engage in any battle in spite having one of the greatest navies in Mediterranean.
