[走る] in this context refers to [ある方向に傾く]. The direction(方向) here is ノルマ(norma in Russian) which means quota. The best translation for the text should take account of the [ノルマ的な] that comes before [出願に走る]. Based on these reasons, I would suggest rush to (meet) quota-based application.
The reason I include meet in my translation is because the sentence structure in Japanese often omits certain words that are deem unnecessary or repetition of words already present in text.
In this case [走る] can loosely be translated into [run] which is a time related verb. The original text can also read as
企業がノルマ的な出願に間に合うように走る。