airline ticket director/employee for business purposes

Yes they  can

from e commerce agents even?? ,

Sec16 [4] A registered person shall not be entitled to take input tax credit in respect of any invoice or debit note for supply of goods or services or both after the due date of furnishing of the return under Section 39 for the month of September following the end of financial year to which such invoice or invoice relating to such debit note pertains or furnishing of the relevant annual return, whichever is earlier.

Yes they can take the ITC even if supplied by e-commerce.

while ticket booking it will be in person's name then how gst number to be updated

From what I have seen till now, MOST online booking websites will either:-

a) ask you to provide GST details while booking (look thorugh the booking page carefully so you don't miss it) - like at IRCTC, MakeMyTrip, etc.  Or,

b) require you to register first, before you do booking (you are supposed to provide your GST details while registering, or at the time of converting your personal user profile to business/corporate profile), like at goibibo.

Such agenst will forward your GST details to the main transport operator / hotel during online booking. So, you can receive a separate GST invoice from them for actual travel/accomodation cost.  Therefore, the GST invoice that you receive from online booking agents is to account for agency charges only.

Note: Sadly, you may have to do some follow-up to get the GST invoice for actual travel/acommodation cost.