This is an expression mentioned towards the end of Ziyarat 'Ashura`, and wurūd here means entry, or returning back. The wurūd is entering into the sovereignty of God, and it could be positive and good, or negative and bad [as in Surah Hud, 98]. Yawm al-wurūd may not just be Judgment Day, but any stage after death, like entering the grave, barzakh, and resurrection. It is during these stages that we ask Allah ta'ala to grant us the shafa'ah of Imam Husain (a.s.).
This is why in the Ziyarah, we say:
اَللَّهُمَّ اُرْزُقْنِي شَفَاعَةَ اَلْحُسَيْنِ يَوْمَ اَلْوُرُودِ
We hope to be blessed with the intercession of Imam Husain (a.s.) at these very difficult times once we depart this world. Imam Husain (a.s.) is the Ark of Salvation, and it is for this reason that we seek his intercession.
Of course, it is more probable that yawm al-wurūd is referring to Judgment Day alone, as such has been expressed in other narrations, like the following in reference to Imam Ali (a.s.) being in charge of supplying water (al-sāqī) on Judgment Day, as the Prophet (s.a.w.) had proclaimed:
اَلسَّاقِيَ يَوْمَ اَلْوُرُودِ
In any case, this ziyarah is an important ziyarah that we should recite as much as possible, and hopefully be granted the intervention of Ahlul Bayt (a.s.) on Judgment Day, at the time we will need it most.
With prayers for your success.