Firstly, by 'SB' do you mean Stormblood or Shadowbringers? (the former is usually abbreviated as 'SB', the later as 'SHB' to differentiate them). If you're meaning Shadowbringers content, then the answer is yes you can go through most of the MSQ without needing to party with other players... buuuut.... there is a catch. Dungeons can use the new Trust system to make up your party (and are reccomended for those who want to experience the story of the dungeon, as the NPCs have their own little story-specific dialogues during them). The three Trial battles that are also part of the MSQ however (
Titania, Innocence and Hades
), require a party of 8 players to complete, you cannot use Trusts with them I'm afraid, and you cannot avoid them either.
Of these three, only the first () gave me the most trouble, several parties I was in were useless and couldn't meet the DPS check, but I eventually found one party that knew what to do and did it without fuss.
Hades, the final battle for the MSQ
, I did have one wipe and I failed the QTE that was a part of it, but we otherwise pulled through with not to much drama; while
Innocence gave... no trouble whatsoever, I was most surprised it went by so easily.
So while most of SHB's story duties you can solo with Trusts, the three Trial battles required at different parts of the story will require you to party with others. There is no way around this I'm afraid.