Is it permissible to exploit personal connections with influential people to obtain visas for Hajj or Umrah, including what violates established regulations, such as using private cars instead of the buses designated for that purpose?

It appears there is no objection to the brother interceding to obtain Hajj visas for himself or others through legitimate means, without deceit. This is because intercession is of two types: permissible, which is good intercession that helps someone achieve their right without injustice, as the Prophet, peace and blessings be upon him, said: "Intercede, and you will be rewarded; Allah will decree through the tongue of His Prophet what He wills." The second type is impermissible, which involves an unlawful matter or injustice. Both types are encompassed in Allah's Almighty قول: (Whoever intercedes with a good intercession will have a share of it, and whoever intercedes with an evil intercession will have a share of it. And Allah is over all things a Maintainer.) What was mentioned is closer to the first, permissible type.