1. we would like to confirm with you the inspection date is for 21 April
2. we would like to confirm with you the inspection date is on 21 April
Can I use 'for' with the date?
Yes, both are okay. It's sloppy, and leaves out some words which would remove your confusion, because they are understood [by implication] to an English speaker.
What is really being said is: "We would like to confirm with you that the inspection date is scheduled for 21 April."
The second one is saying: "We would like to confirm with you that the inspection will be on 21 April."
The difference is subtle. The second implying that the date is set, and not alterable. The first suggests that perhaps, if you contact the inspector, the date can be changed, as it has been "scheduled" but not necessarily confirmed.