Is it permissible to travel to from Doha to Jeddah without tying the ihram (as we plan on going to Madina first)? I’m concerned about entering in the boundaries of Mecca without ihram.

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Yes, you can perform Ihram through a vow from Jeddah airport, according to the fatwa of His Eminence Sayyid al-Sistani.
However, since you are going to Medina, you do not have to perform ihram from Jeddah, rather you perform it from the meeqaat of Medina, which is Al-Shajarah Mosque (Dhu Al-Hulayfah).