There is a tendency in some quarters to insist that “they didn’t do that, therefore they wouldn’t and couldn’t do that”, particularly amongst people who haven’t bothered to check into the details of a subject; hence the bulk of the dogpile of comments above.
Your question is a good one and by no means out of scope of either the requirements or capabilities of the Kreigsmarine, nor in fact far beyond what they did indeed do. It is firstly important to consider the German Navy’s doctrine and mission, and how a light carrier would have conformed to, and enhanced, that mission.
The capital ships of the Kriegsmarine’s surface fleet were not intended to form a battle fleet to challenge the Royal Navy’s control of the Atlantic; such a mission was recognised in 1935 as being well beyond the scope of the KM for the foreseeable future. The 1939 plan for the expansion of the Kriegsmarine (Plan Z), which was for expansion of the fleet to the point where it could challenge the Royal Navy in battle, wasn’t expected to be completed until 1948, none of the major surface vessels proposed in Plan Z were completed, most weren’t even laid down, so we need not consider that further. The role anticipated for the Kriegsmarine prior to 1939, which was its primary role during World War Two, was to interdict Britain’s merchant fleet and force Britain to the negotiation table by disrupting British maritime trade and sinking cargo ships. All of Germany’s warships were designed around this mission; even the Bismarck, despite being the most powerful battleship afloat, was first and foremost a commerce raider. This was KreuzerKrieg: Cruiser Warfare.
The principal problem of cruiser warfare was simply finding a ship to attack in the enormous expanse of the Atlantic Ocean to begin with. This was encountered by the Germans in the First World War, and the building program of the 1930s included efforts to cope with this. Prior to 1935, the German navy was prohibited from building aircraft carriers, and the warships it was permitted were restricted to 10,000 ton displacement; despite this size restriction, the Deutchland-class cruisers (generally referred to as "pocket battleships") each carried 2 x Ar 196 floatplanes, which represents a considerable allocation of resources considering the small size of the ship. Prior to 1935, the Germans also gained experience operating aircraft as sea through the use of Lufthansa catapult vessels. These were ships that had large catapult on the aft deck to launch two and three-engine flying boats; generally the Dornier Do 18. So clearly the requirement for a maritime aviation was both recognised and given a reasonably high priority. Nor is this surprising, since the North Atlantic is four times larger than Europe from the Urals to the French coast: reconnaissance was indispensable.
In 1935 the Baldwin Government, in an effort to avoid a naval arms race with Germany, negotiated and signed the Anglo-German Naval Treaty in 1935. This “restricted” Germany to a fleet 1/3 the tonnage of the Royal Navy, including aircraft carriers and submarines. (A provision of the treaty even permitted Germany to build up their submarine fleet to match that of Britain "should they so desire".) In fact the treaty was no restriction at all; Germany’s shipyards would have been fully occupied for ten years turning out a fleet equal to a third of the Royal Navy, and would have tied up enormous amounts of German resources; all of the ships completed between 1935 and the end of World War Two came under the Anglo-German Naval Treaty. The treaty simply gave official British approval for Germany to tear up the naval provisions of the Treaty of Versailles.
The importance of naval aviation for reconnaissance by this time is reflected in the fact that the new Hipper-class heavy cruisers and the battleships Gneisenau and Scharnhorst all carried 3 x Ardo Ar 196 floatplanes, armed reconnaissance aircraft that could vastly increase the frontage of ocean that the battleship could sweep in its search for prey. The larger Bismarck and Tirpitz both carried 4 x Ar 196 reconnaissance aircraft. In addition the Kriegsmarine built and operated a string of seaplane tenders; Krischan I (commissioned in 1934), Krischan II and Krischan III (’35), Grief (’37), Hans Rolshoven (’38), Karl Mayer (’40), Max Stinsky (’41), Immelmann (’41), Drache (’41) and Boelcke (’42). These ships launched the Do 18, DV 138 reconnaissance flying-boats and the He 114 floatplane. Finally there was the aircraft carrier Graf Zeppelin, laid down in 1936 and launching it in 1938. Given the small overall size of the Kriegsmarine, clearly a significant proportion of it was in fact involved the use of maritime aviation.
The Graf Zeppelin was 85% complete when work on it was halted at the end of April 1940; priority being given to other vessels. Graf Zeppelin displaced 33,500 tons, making it slightly larger than the Scharnhorst-class battleships that it was intended to accompany on commerce raiding missions; by comparison, HMS Ark Royal displaced 22,000 tons.
Clearly, for Germany to have had an operational aircraft carrier in World War Two, it would have had to be completed prior to the beginning of the war, so something smaller laid down in 1936 instead of Graf Zeppelin. Optimally this would be a vessel of the Hipper-class in size, and built using the keel of the Prinz Eugen or Seydlitz. At 17,000 tons, this would be slightly larger than the Royal Navy’s HMS Hermes, so a complement of 20 aircraft would be reasonable.
For aircraft, the Me 109 would have been out; not only would its undercarriage have been far too unstable and easily damaged for deck landings, but its priority as the front-line Luftwaffe fighter would have made it unavailable anyway. Fortunately there was an aircraft available which did suit the requirements of the Kriegsmarine for a carrier-borne fighter: the Heinkel He 51. (At the start of the war, the best FAA fighter was the Gloster Sea Gladiator, so for naval aviation purposes, the otherwise obsolete biplane fighter was still valuable.) The durable undercarriage of the Heinkel, as well as its slower landing speed, would have made it quite desirable for carrier operations in the early war period, later the Fw 190 would have been desirable, but hard to gain priority for. In the reconnaissance/torpedo bomber role, the Fieseler Fi 167 was available, this was capable of carrying a torpedo of similar size to the Fairey Swordfish, but had both greater speed and greater range.
The potential of such a vessel, accompanying the Gneisenau and Scharnhorst in their hunt for convoys, would have been considerable, not to mention accompanying the Bismarck. In fact the threat posed from even a light carrier accompanying a battleship would have greatly disrupted British convoying; battleships began to be used to escort the convoys in 1941 because of the threat posed by the German battleships, but a battleship, rather than being able to provide protection to a convoy, would itself become a major target to torpedo bombers.
