I received these two trees (a juniper and a maple) around July last year, and I've pretty much done nothing but just water them to try to keep them alive the first year. Only thing I've noticed is that the soil gets very dense and wet at times, so I thought it would be a good idea to repot. Is repotting right now a good idea? If so, any specific soil I should use for these two? Also anything else I should do with these trees at the moment?

yes, it's possible, where do you live and what kind of land do you have for repotted

I live in the United States, Zone 7a. I was thinking of just repotting in the same pot they are in right now, just replacing the soil to a more inorganic one.

Yes you can repot into same pot.

I tend to use a large extend of organic soil for trees in nurcery pots. Maybe 20% inorganic for better drainage. I save bonsai soil for bonsais in bonsai pots.

I wanted to do a more inorganic mix because last year, I had trouble with the soil being way too dense and wet even with moderate water, and especially during rain. Is this alright to do, or should I avoid an inorganic soil?

You can use any soil you prefere. My 20% inorganic is more than most nurcery plants have. Some people use 30,40,50 all the way up to 100% and everything inbetween. Everyone have their own mix and most swear on it as the best. The important thing is to adjust watering and fertilizing to your substrate and climate.

Start off thinking mostly inorganic, but add organics, possibly a lot, to hold water, depending on your watering habits and ability.

If you check them two or three times a day, watering when necessary, completely inorganic should be fine. If you life allows you only once a day, you might need the water retention you get with organics in some months, like August. I am in 8a and water twice a day during most of August and some of July and September.

You could get a cheap houseplant or pot with some kine of ground cover, put it out with the trees, and check for when it starts to wilt. Then, water everything.