What is the ruling of Islamic law regarding the establishment of a society to help the poor and needy perform the obligation of Hajj, and is it a condition to pay the costs of Hajj from one's private funds?

The work you have undertaken — collecting money to enable the poor to perform Hajj and Umrah — is a good deed for which the doer will be rewarded. Whoever claims that it is impermissible has erred. There is no harm in collecting money for this purpose, as it is among the acts of righteousness for which those who participate with their money or effort will be rewarded.