If possible, run the side session on another time or day with just those participating
We've actually done this a number of times in a campaign I'm playing in. The players or DM occasionally come up with side quests relevant to a select few players, who we then find a time to play through an another day of the week. This prevents it from disrupting the group as a whole by keeping them from playing, while allowing for the side-quest to still take place.
The smaller group also helps the interaction go much faster. One session was literally the DM and I on a 15 minute car ride, playing in the theater of the mind. Another one was just my coming along to assist on another player's side quest, and only took us half an hour. Yet another of mine took about an hour, and was very similar to what you described above (me, the rogue, on a secret thieving quest).
So yes, this is entirely possible and normal. As I mentioned above, just do your best to avoid disrupting the other players, either by making them sit through it, or by making them have to skip what would have normally been a group session altogether.
Be aware though that by allowing this for one character, it may open up the door for others to request the same. This doesn't mean you have to oblige, especially if you only intended this to be a one-off opportunity for help the other player connect to their character better. But be prepared to know how you intend to respond and handle such requests.
Impact to the main story
This part is a little more dependent on you as the DM, and how you want this to affect your world. In the same way that backstories can (and should) affect the overall plot to increase character engagement, so can and should present stories. Ideally you might want to avoid consequences that take agency away from other players in their part of the story, but that doesn't mean it can't be relevant.
For example, in my thieving side quest I mentioned above, completion of the task got me into a city-wide underground criminal network. This allowed me to help other members of my party as an informant, who could get information on quests and goals we as the whole party sought, and helped increase our group dynamic. This had an affect on the story by helping to improve my investment in it and provided another resource for the other players, but didn't take anything away from the others.
Side-note: Handling XP
As NautArch suggested in a comment below, one thing to keep in mind is how this might affect XP gain. Depending on what method of leveling you use (standard XP, milestone, 3 pillar, etc), you will want to be careful that these side quests don't give someone an unfair lead over their fellow players. In our campaign, side quests typically only granted XP to those who were behind the others, often with the quest itself being an intentional way for that player to catch up.
As someone who typically attended every session and was at the lead in XP, I would often agree with my DM to forgo it altogether in order to simply enjoy the adventure, or in other cases would join on someone else's side quest (if they were ok with it) as a way to help them level their character faster. Again, this will all depend on your group and how leveling works, but so long as the side quests aren't giving them an unfair/special advantage, you should be good.