Today is the first day after summer vacation for the hospital so it's quite crowded.
ご質問ありがとうございます。
「今日は病院が夏休み明け初日だったので混んでいた」は英語で「Today is the first day after summer vacation for the hospital so it's quite crowded.」と言います。
もし「[患者](https://eikaiwa.dmm.com/uknow/questions/36630/)さんが多くて」も追加したい場合、そうすると「Today is the first day after summer vacation for the hospital so it's quite crowded with patients.」と言えます。
ご参考になれば幸いです。
The hospital I usually go to was so full of patients today because it was the first day they were open after their summer holidays.
この場合、次のような言い方ができますよ。
ーThe hospital I usually go to was so full of patients today because it was the first day they were open after their summer holidays.
「普段行く病院は夏休み明け初日だったので、今日は患者さんでいっぱいだった」
the first day they were open after their summer holidays で「（病院の）夏休み明け初日」と言えます。
ご参考まで！
アメリカ合衆国 
カナダ 
