'copy the whole record into another worksheet'
I interpreted whole record as whole row.
Put the name of 'another worksheet' on code line 5 (=Sheet2 now)
You also didn't specify the column where the EID's are located. Adjust the column letter in code line 6.
Apply changes and give the following code a try:
Dim fValue As Range
Dim sColumn, ShName As String
ShName = "Sheet2"
sColumn = "A"
svalue = InputBox("What is the EID to look for?:")
Set fValue = Columns(sColumn).Find(svalue)
If fValue Is Nothing Then
MsgBox "No EID found"
Sheets(ShName).Range("A" & Rows.Count).End(xlUp).Offset(1, 0)