Will the B constant for 10k ohm NTC thermistors be the same from all vendors?
No, yes and yes, in that order.

B values vary a lot, typically between 3000 K and over 5000 K, not only between manufacturers but between different parts from the same manufacturer. Even apparently identical parts from one manufacturer will be subject to a tolerance on their B values.

You can measure the resistance of the thermistor at various temperatures and plot the log of resistance against reciprocal absolute temperature, ie ln(R) against 1/Tabs.

The slope of the best-fit straight line through the points will be an approximation of the B value (which as Olin notes is itself an approximation of the Steinhart-Hart cubic, itself an approximation ...) .

For the benefit of those who have not heard of the B (or \$\beta\$) parameter, there are explanations on Wikipedia and on this manufacturer's website (the first I found with a google search).