Smaller vendors are more likely to build and deliver to customers' requirements because the decision-making team is likely to be hands-on, very technical, and involved every step of the way.
Fast decision making, ability to adapt to changes faster, accuracy is more likely to be found across smaller vendors because they are more nimble, agile, faster and are likely to be a more cohesive/non siloed team.
Amazon & Google all started off as startups with smaller teams, their founders were all heavily involved (hands-on) at the start, which will explain their very successful kick offs. Not so sure about right now.
Smaller vendors have more to lose in terms of litigation, loss of reputation, face bankruptcy, and will do whatever needs to be done to deliver all their projects successfully.
So, next time you are awarding a Software Project or an IT Project to a vendor think about it carefully as the decision to go with a big name may not necessarily get you your result.