@Jukra thank you for your answer. My thinking was roughly the same, with a different procurement decisions during the Interwar period.
One thing that would certainly be helpful is modernization and standardization of the artillery, with Finns asking for French help in that regard. French would have a sizeable amount of various artillery weapons availlable, and Finns could really benefit from standardizing on French models, instead of mish-mash they had to make do IOTL. They would also receive some of the French artillery doctrine along with the guns, and mindset of materiel over men could perhaps ensure that there are enough shells acquired and produced from domestic sources, so that Finnish artillery is able to do counter-battery and saturation bombardments more often then IOTL.
Finnish infantry did extremely well in OTL, and I am not really certain what could be done there that would further increase their effectiveness. Their basic infantry weapons were adequate for the task, and while rimmed cartridges are often derided, I really do not think that they should try and change cartridges, that would really eat into their military budget substantially, for little actual gain. Perhaps starting the production of their M39 Mosin variant somewhat earlier would be useful, if only to create a reserve of arms in case they are needed, with older Mosin variants stockpiled in case of war? M31 SMG is a nice weapon, only thing needed there is increasing production as much as possible, since type of combat really favoured the SMG. One thing that their infantry would really benefit from is having a decent LMG availlable in larger numbers. Their Lahti-Saloranta M/26 was a rather complicated weapon, with 188 individual parts, and it seems it was rather difficult to manufacture, so a different weapon may be preferable instead. Czechoslovakian Vz.26 LMG may be a good choice, if they can quickly modify it to use rimmed ammunition, or they could somehow acquire examples of Soviet DP-27 LMG and produce that instead.
You raise a good point with AT weapons, both guns and individual. AT guns are perhaps the easiest, there were numerous light AT guns availlable during Interwar period in Europe, since many of the nations produced their own. Unless I am mistaken I do think Finland did produce Bofors 36mm AT gun under licence, but their domestic only started in 1939, and they had roughly 90 or so of these guns at the beginning of the war, either bought or made domestically. If they make a decision to acquire a licence/start production earlier then OTL, then we could see larger number of rather good AT guns in Finnish hands, then what was the case IOTL.
As for the individual AT weapons, one can assume that Molotov Coctail will make its appereance, as will the various bundled grenades and satchel charges. Have engineers put together a bunker busting satchel charge, and infantry takes notice and adopts it as an AT weapon of sorts. That is the best I think they would have, AT rifles were relatively popular, and Finns did have their own 20mm version, but that would be more or less it for the Winter war, no HEAT or anything like that, just existing weapons, but in greater numbers then IOTL.
Some motorization could perhaps be done, not in the sense of having motorized divisions, but with logistics, medical and other support elements being motorized to a certain degree, so that horses do not have to be taken from civilian economy. It would also reduce the number of people required to take care of the horses, as well as reduce the fodder required. Some of the artillery, especially of the heavier variety could also be motorized, allowing them to become much more mobile then they would be otherwise. Most, if not all of these trucks would have to be imported, since I do not really see Finland getting a motor industry capable of churning out thousands of cars/trucks/tractors needed for their civilian and military needs. Finnish government could subsidize the motor vehicles, so that they subsidize someone to buy a truck, but for a period of years the government has the right to take that vehicle in case of war/emergency, that would at least partialy help.
Thoughts?