I want to replace my cassette which is 7 speed 14-34 shimano. The problem is I can't find exactly the same one. All I saw its 13-34 or 14-32 cassettes. So the question is: It is possible to replace my cassette with 14-32 or even lower e.g. 13-26 without replacing a chain or any other replacements? Thank you in advance.
As long as you're not adding larger cogs than you've got on right now, your existing chain should be a functional length.
One more thing to consider is that if your current chain has significant wear, running it with a new cassette can damage the cassette (effectively very rapidly turning your "new" cassette into your "old" cassette), so even though your old chain should work fine with your new cassette, it may be worth it to replace it anyway.
If you're replacing it with a similar range (1-2 teeth at either end) you should be fine. I wouldn't recommend changing the gear range significantly.
I replaced my cassette with a much smaller biggest (in terms of number of teeth) gear and it wouldn't shift into the top couple of gears. I believe the length of the pivot arm on your derailleur is tuned (to a certain degree) to the range of your cassette.
I agree with prototoast that you should probably replace your chain at the same time.