They didn't need that, they were simply creating vassal states with local rulers, without much further intervention in internal affairs. Some vassal states used Mongols' military power to their advantage, against neighbors. Especially when they didn't have much choice, the alternative was to be invaded and robbed.
Oh, vassals...
Like the Duke of Toulouse was a vassal of the king of France, where the 13th c. political trobadour Peir Cardenal could publicly exhort the good Duke to take up arms against the Occitans' traditional enemies, "li frances e li masmut", the French and the Mohammedans? They did too...
Not to mention the kind of utter mess that was the HRE. Sure, the Mongols might get the Lombard League as vassals. It would likely profit them about as much as they did the Emperor i.e. constant state of war against the citzen-soldiery of the aggresively independent north Italian city states led by Milan. If the Khan wants compliance, he needs to go and seriously enforce it... For as long is the bloody example of his latest effort would last...
Besides, for instance, try telling Ecclesiastical Potentates like the German warrior-bishops of affluent, fortified, trade centers places like the bishoprics of Mainz, Paderborn, Minden etc., that the Khan had replaced the Emperor, and just watch the non-compliance manifest itself. (Now try it with the Khan having replaced the Pope, which would be a lot more interesting, but begs the question how that would happen?

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Venice was a vassal of Constantinopolis. Didn't bother the Venetians any when invading the Byzantines. Half the time the Venetians were also excommunicated by the Pope for repeated shenanigans, which typically engendered a classic respose by one of its diplomats: "We're just as good Catholics as the Pope."
So, if the Mongols would be content with that kind of symbolic-de-facto-independent-vassals, sure...
The impression is however that the Mongols had something rather more compliant and hands-on in mind. In which case western Europe would likely provide an endless source of trouble, renewed conquest and re-pacification, probably for very little gain. You need to be on site, in charge, and on top to actully control local conditions in most of Medieval western Europe. Fail to do that, and overlordship was likely as not to quite often be stunningly symbolical in nature.
At least it requires the Mongols recruiting very considerable numbers of local clients, whom they need to offer very tangible incentives, and from which they likely can expect very little in return beyond lip-service. Nothing personal re. the Mongols, it's just Medieval western Europe being a bit of a mess with respect to autocratic chains of command, where even Mongols looking for subservience might have to resort to acknowledging the rights and privileges of their vassals as per traditional specifications. (The oath taken by the reps of the city of Barcelona when accepting a new king as overlord is a classic it's along the lines of we who are no worse than you, accept you, who is no better than we, as king, provided you respect ALL our freedoms and privileges, and if you don't, we won't.)
What the Mongols likely could not do would be to use such clients to transform the European political landscape. Done like that the Mongols might perhaps add some kind of symbolic overlayer, but it wouldn't touch fundamentals. It could be something along the lines of Europeans formally recognising Mongol overlordship, on the premise that the Mongols just leave it the hell alone. It would work on the premise that a Mongol Khan somewhere could be the symbolic overlord just as well as any locally absent king.
If the Mongols would dream of actually controlling things, it's invasion and endless sieges of inumerable small fortfied strongpoints that needed to go on the agenda. They would have to bludgeon western Europe into submission locally on a case-by-case basis. That could force a radical change in western European societies and polities for sure, but it would most likely need precisely something as radical as that to make it stick.
It seems to me what's at stake is the meaning we put into concepts like "conquest" and "control"?
So, vassal states? Sure. Provided the Mongols play by the European book. Should they harbour any great hopes of the western European polities realising they should be just generally subject and subservient in a less peculiar fashion at least without requiring Mongol presence, and repeated breaking of heads then they would likely be disappointed.
Which amounts to saying that in order to create these vassals, the Mongols would need to invade, and then confront the state of devolved, local political power that most of western Europe ran on. I'm really not saying they absolutely couldn't do it. I do however think some of the realties of European politics might cause a bit of consternation. They could of course go for just literally slicing through it to achieve some other state. But while it would require considerable effort to just defeat western Europe militarily, it would likely take even more effort to confront, and substantially change, the social and political realities of the place.
Which brings us back to the problem of "holding". Certainly some Mongol ruler could set himself up with a bunch of typically relatively unruly western European vassals, should said Mongol decide to deal with the Europeans like Europeans conversion to Christianity and a position as another Christian prince would help. A Mongol Khan ruling western Europe as undisputed Khan, with no concession to the western way of doing things, at least as far as his comportment in that part of his realm was concerned? Now that would likely take some very serious effort.