I want to know if we have "postponer" in English or not? Microsoft Word underlines this word as incorrect word; but when I search in Google, some online dictionaries have this word.
Can I use this word in English?
As you note, postponer is in some dictionaries, such as the Oxford Dictionary of English.
However, I don't imagine that most people know the word. We all have mental dictionaries with a set of words and phrases, their meanings, and how they're used, and for most people postponer probably isn't in this dictionary. It's simply not common enough yet. Damkerng's suggestion of procrastinator is more well known.
That said, the suffix -er is productive, so you can derive postponer from postpone. And anyone you're talking to can do the same, so they'll be able to understand it just fine, assuming it makes sense in context. (Unfortunately, since we have no context here, we can't tell whether it makes sense or is appropriate.)
Since it's not an established derivation, it's possible some people will think it sounds funny or informal. Whether it's appropriate in a particular context is a judgment you'll have to make.
Microsoft Word's spell checker compares what you write to a dictionary. No spell checker will know every word you use; if you know a word is legitimate, ignore the red underline and move on. Technology has its limits.
One of English's virtues is that it is very malleable. Or adaptable. If you follow the few rules about prefixes and suffixes, you can put things together however you like and have it be 'valid' English. See Antidisestablishmentarianism. Out of 28 letters, only 9 come from the root word. The rest of the letters come from prefixes and suffixes. You can even take nouns and use them as verb, if you so choose. i.e: "Did you Google it?"
With that in mind, saying that someone who postpones is a postponer seems wholly correct.
That said, I have to disagree with what was said here. Procrastinate is almost certainly not the word you want to use instead of postponer. Procrastinate means to "put off doing something, especially out of habitual carelessness or laziness". Someone who does that is a procrastinator.
On the other hand, to postpone means to "delay until a future time; put off".
So for example: Imagine it was raining during a soccer game and it is your job decide if there was too much rain. If you decided to call the game and reschedule it for a later date, you would be postponing the game. You would be a postponer. But, in no way, shape, or form would you be procrastinating. The "habitual" and "laziness" aspects are inseparable from procrastinating. But postpone doesn't have any connotations like that at all.