1

Today, I came to office without Id card and hence took temporary staff pass. I just need to inform my HR on this by email which sentence I should use out of these two.

I have forgotten my id card today.

or

I forgot my id card today.

2

The first version is more clear since it shows that you still/currently have the situation of your ID card being not available (= you have it forgotten somewhere). The version with Past Indefinite just describes past events: who knows, maybe you've already solved that problem? However, in the given context (asking for a temporary pass) both versions work.

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