Sammy wasn't free from the injury bug. Let's start with 1992, his aforementioned first year with the Cubs. He hit 10 home runs that year, but he 33 home runs in 93 and 15 home runs in 90 (he spent signficant time in the minors in 91), so if he hadn't been injured it's safe to say he would've hit at least 20 home runs in 1992. Add 12 home runs to his total, giving him 621.
Next we have 1996. Sammy played only 124 games this year, missing the last two months due to injury. Yet he still hit 40 home runs this year. It would have been very possible for him to hit a clean 60 home runs that year, but that is asking a lot and Sammy was very streaky as a young player, so let's be conservative and estimate he hits only 50 home runs if he plays the whole season. That makes his total 631.
Next we have 2003. Sammy some time on the DL and served a short suspension for #PoppingTheCork; he ended up hitting 40 home runs in only 137 games. Since Sammy took more games to hit 40 home runs than he did in 1996, we can't assume Sammy would have hit 50 if healthy, so let's assume he finishes a healthy 2003 with 45 home runs, making his career total 636.
He was injured even more in 2004, most notably because of violent sneezing but he had other injuries as well. He played only 126 games and finished the year with 35 homers. He played about the same amount of games as he did in 96 and hit fewer homers, so if we're using that as the standard like we did for a hypothetical 03 do-ever, I'd say a healthy Sammy would have finished 2004 with 45 home runs again, since Sammy was selling out for power like crazy this year. That takes his total to 641.
He had a shitload of injuries in 2005 with the Orioles. He played only 102 home runs and finished the year with only 14 home runs. What people forget about this season his that he started it pretty decently before his injuries. Sure, his power mysteriously went away, but he was hitting at or near .300 for most of April, then he slumped in the first week of May before sitting out the rest of the month with an injury. He was never the same after coming back from that injury, and then he ended up missing the last two months of the season anyway to add insult to injury. So yeah, his power numbers were down in that first month, but he was making contact and putting the ball in play pretty consistently and this was in a great hitters' ballpark, so it's safe to say he would have started hitting home runs eventually if he played a healthy season. That's not taking away from the power slump to start the season, but let's assume he finishes the year with a serviceable 24 home runs, making his career total 651.
I won't add a hypothetical 2006 season, nor will I mess with his 07 numbers because 1) he spent no time on the DL and 2) ending his career with 21 home runs is too poetic.
So yeah, a healthier Sammy Sosa would have hypothetically hit around 651 home runs, conservatively. I don't know where that would put him in the all time listing if everyone is assumed to be healthy — Bonds would certainly be an 800 home run guy, and Junior and A-Rod would be well into the 700s, to say nothing of Ted Williams — but I'm also not sure if I want to live in a world where Sammy Sosa doesn't try to make "Mr. 609" a thing on social media. Mr 651 just doesn't have the same ring to it.