Probably the first armored vehicle to serve in Finnish army was the American Peerless armored car, captured from the Soviets in 1919 by the Finnish. This 7-ton armored car was one of the 16 vehicles first bought in 1916. I haven't found any real details about this vehicle, but it probably looked something like this (this is a later model though):
This vehicle was captured by Estonian volunteers however and after the civil war ended, it remained in Estonia. It was called "Pohjan Poika" by the Finnish ("Son of the north"). Another armored car from this era, captured (some sources say bought) from the Russians was the Fiat A armored car. Four were obtained during the turmoil preceding the Finnish civil war in 1915 and they continued to serve in Finland until early 20's.
The third armored car to make it to Finland was the Soviet Austin-Putilov armored car. Basically, these armored cars consisted of the Austin chassis (during the war, that was all the British factories could provide) with Russian modifications. This armored car was the most important Russian armored car of WW1.It was equipped with two machineguns and armored with 8mm thick plates. Apart from the Russian army, it also served in the Japanese and Polish forces. The Finnish captured a few in 1918 and used them until the 20's.
But those were just the armored cars. The first real tanks were purchased by the Finnish in 1919. Those were the ubiquitous Renault FT-17 tanks. 14 were purchased with the 37mm Puteaux gun and 18 with the 8mm Hotchkiss MG. Two of them actually saw combat: they served with the "white" troops of general Yudenich against the bolsheviks. After their defeat (Yudenich himself escaped capture and fled to France, where he lived until his death in 1933), they were returned to Finland, but they were in such a poor shape that France supplied to additional vehicles.
Finnish FT-17's continued to server deep into the 30's. By that time they were totally obsolete, but high costs of modern armor prevented Finnish to buy anything better and the FT-17's formed the backbone of Finnish armored corps until 1938. Even in 1942 a few were still used for training, others were turned into pillboxes in Winter War.
Of course, the Finnish army experimented with other types of vehicles too. St.Chamond M21 Chenilette wheel-cum-track design was purchased (allegedly in several exemplars), but the design was found unsatisfactory and was rejected. Another such vehicle was the Vickers Mk.IVB tankette, Vickers Mod.1931 amphibious tank, Vickers Mod.1933 or the famous Vickers 6ton. Here, sources differ about their use. The St.Chamond disappeared from the scene really quick, but the 1931 and 1933 light tanks were used for training until 1939 or even 1942.
The only tank from that period, purchased in larger numbers was the Vickers 6ton, model E, type B. The Finnish called it "Viku". The first prototype, ordered from Britain was shipped without machineguns with only a 47mm Barrow gun. 16 more (this time with no armament at all!) were shipped in 1938 and another 16 in 1939. Since these tanks lacked their own guns, they were refitted with the FT-17 37mm Puteaux guns and Hotchkiss machineguns from the same source. This too was deemed unsatisfactory in 1939 and it was decided to re-arm the tanks with 37mm Bofors AT guns (or rather local copies under the designation 37 psvk 36) and with turret-mounted L-33/36 machuneguns. The hull machineguns were removed and instead 9mm Suomi submachineguns were installed. These vehicles fought in 1940 against Russians near Honkaniemi in February 1940. The Finnish crews - unexperienced at the time - suffered losses with 8 tanks being lost due to breakdowns and combat.
Since the Finnish captured a large number of T-26 tanks in the war with the Soviets, it was decided to rearm the Vickers tanks with the Russian 45mm guns and DT machineguns. Further modifications were made (including the modifications to drivetrain and suspension) and such modified tanks with Soviet guns were redesigned to T-26E. These vehicles could be recognized from other captured T-26 tanks only by careful detail observation and they were so well regarded they stayed in service until 1959.
Another vehicle worth noting that was purchased in this period would be the Swedish Landsverk L182 armored car. It wasn't really a success and it was later transferred to Finnish police. The Finnish however did capture a number of BA-20 and BA-20M Soviet armored cars (as many as 20), they were kept around as scouting and liaison vehicles until 1951. Their official designation in the Finnish army was BAB, but they were nicknamed "Beibi".
Heavier armored cars (such as the BA-6 and BA-10) were also captured. They underwent a refit, where their original Ford IV 50hp engine was replaced with the stronger Ford V8. These stayed in service until 1959, just like the T-26E tanks.
Source:
Finská obrněná technika 1917-1945
http://mailer.fsu.edu/~akirk/tanks/