、。
She got sick on her day off and had to find a doctor.
Literal
She [topic-は] day-off [on-に] sick became, doctor [object-を] had-to-search.
Two clauses linked by 連用中止法 ('masu-stem suspension'): 病気になり (the masu-stem of 病気になる) replaces the te-form 病気になって as a written-style way of chaining clauses. The form is more formal and more written — you'll see it in news, essays, and literary prose, but rarely in everyday speech. ~なければならない is the standard 'must, have to' construction; the past form ~なければならなかった carries the obligation back into the past. The phrase 医者を探す (rather than 医者を呼ぶ 'call a doctor' or 病院に行く 'go to a hospital') suggests the location wasn't familiar, hinting she may have been somewhere away from her usual healthcare.