Depending on definition, Lapland covers almost half of Finland so obviously the weather can differ a lot depending on location.
I tried to find some weather sites with historical data in English. This site has few cities you can browse and the data seemed accurate compared to other sources:
Weather Rovaniemi – Monthly Weather History| freemeteo.fi
I'd say make extra preparation to have a change of clothes, including boots/socks and gloves/mittens in case it's raining.
And reindeer cannot be legally stalked, it's considered domestic animal. There's some licenses for wild reindeer / forest reindeer / Finnish forest reindeer (in Finnish: metsäpeura, directish translation forest deer) but they're kind of scarce.