'I look forward to your reply' is a standard formula used traditionally in more formal correspondences, particularly business and other official letters. It tends to be written in its full form, and isn't usually shortened. It's not wrong to shorten it to 'Look forward to..', but this form is more unusual. These days, especially in emails, it is fine to use the more everyday statement 'I'm looking forward to..'. As this is a more informal phrase, it is also fine to abbreviate it to 'Looking forward to..', in the same way as you might sign off by saying 'Hoping to hear from you ...' instead of 'I'm hoping.....'.

You can say " I look forward to your reply" or I am looking forward to your reply". You can also say " Look forward to your reply".

'I look forward' is saying that I am looking forward to hearing from you, but just using 'look forward' would sound like you are demanding someone look forward, as in 'look forward when you are driving'. So in answer to your question, you always need to say 'I look forward to', or 'I am looking forward to' in order to be polite.