I mean, sure. If you want to gimp yourself by starting as a variant human rogue and using all three feats on getting more skills, just to unnecessarily overlap with the rest of the party, you can do that. Nobody in their right mind would do all of that, though.
Rogue = 4 skills
Background = 2 skills
Variant Human = 4 skills(human bonus plus the skilled feat)
Using both ASIs = 6 more skills.
And that only gives you 16 out of 18 skills. You'd also need to multiclass two times into ranger and bard and be 10th level.
That is not what I was thinking.
Hexblood - 2 skills (you could also do Dampir, Reborn or Half-Elf, but Hexblood is probably the best race for this build due to the racial spells)
Background - 2 skills
4 levels in Scout Rogue - 6 skills (expertise in 4)
4 levels in Lore Bard - 4 skills (expertise in 2)
Skilled Feat - 3 skills
Skill Expert Feat - 1 skill (expertise in 1)
On point buy at level 8 S8 D16 C10 I12 W13 Ch 18
Sample skill scores at level 9:
Athletics +7
Acrobatics +7
Slight of Hand +7
Stealth +7
Arcana +5
History +5
Investigation +9
Nature +9
Religion +5
Animal Handling +5
Insight +5
Medicine +5
Perception +9
Survival + 9
Deception +12
Intimidation +8
Performance +8
Persuasion +12
You absolutely could do it quicker with a crazy multiclass, but the character above is going to be pretty awesome with spells, cantrips and Rogue levels.
This is a very viable character, and while I have not done exactly that, I have played a half-Elf Scout Rogue with a 2 level Trickery Cleric dip and similar ability scores. That character took the skill expert feat (boosting Dex to 18) and had 11 of 18 skills at level 6 (5 with expertise) and was totally viable as a character.
There are far better feats to take and the group is going to have those other skills covered most likely anyway.
Your whole point was the party of 4 did not have all the skills covered. I covered all of them with 1 character.
The other three members in the party have AT LEAST 12 skills between them, so even if the example character above used no feats at all on skills he would still have 14 of 18 skills covered, 6 with expertise and with the party members covering the rest with quite a bit of overlap.