I understand that Origami paper may not be available everywhere. However, don't worry, there are plenty of other alternatives that you can opt for. For example, Art Paper, card-stock, construction paper, etc. are some easily available papers with sufficient thickness that should help you in getting your origami to perfection.

I have also been in your position and can understand your frustration of not getting the form right. While making a vase, it is always better to opt for a thick paper that has the strength to hold the rather complicated design of different structures. With the cuts and folds that you make, you have to be precise and clean, otherwise you will end up with a rather messy result. So, ensure that you keep your work clean and detailed. As you said you are new to this art form, keep practicing and you will get the hang of it soon. Best of luck. Enjoy your new found hobby.

An empty gel pen is actually a great substitute for scoring paper. I have been looking for a replacement and this is by far one of the best substitutes that I've found and I often recommend it my other friends as well. What a way to make use of the scrap lying in your house.