I'm curious about the copyright laws regarding using content someone else has created for profit.
For example, Vecteezy is a site where designers upload their vector artwork for others to download for free.
Could I use one of those images to help design something like greeting cards or stationary and sell them?
To answer the question given in the title: Every country has their own laws. Some of the limitations are based on international agreements, so approximately the same law can be valid in many countries. Some countries may not have such regulations.
There are many people who do not respect laws at all, but some people do not respect copyright because it's today very easy to replicate in whole or partially pictures, software, music etc. which can be distributed in the web. They steal because they think they live in a place where the copyright owner has very little power to act. But often they for ex. do not print money because even the poorest and most overpopulated country will act fast and violently when that happens.
The ability to want to steal is between one's ears. The thief of intellectual property may explain to himself for ex. "Actually I didn't steal. It was available with no obstacles and all in all I did not take even a single atom of anyone's property."
In USA the asked law is the Copyright Act 1976: https://www.copyright.gov/history/pl94-553.pdf
I'm afraid ordinary people do not want to study the linked text. Fortunately there are lawyers who earn their bread by being able to read, understand and apply texts like this. Different organizations publish practical copyright guidance for their members and users of the creations of the members. You find easily easy to read explanations also by searching for Copyright Act 1976.
Finally the websites which distribute legally creations as files and the distributed files themselves contain the license - the exact conditions and limitations of the usage and redistribution. Copyright laws declare that the copyright owner have a right to state the usage and distribution conditions and limitations and, of course, the price.
The text of the question asked a different thing.You asked can you use an item shown in Vecteezy to help to design something for sale. You asked a legal advice. I do not give it for 2 reasons: I am not a lawyer and I do not know what you mean by writing help to design (see NOTE1). But if you copy a part from someone's work it will be legal only if the license allows it and it also allows the way you are going to distribute your work.
NOTE1: Help to design is quite a loose concept. I assume it would be legal to purchase a nice drawing of flowers, to get into good mood by watching it and then design a new waste bin cleaning machine which contains no images of flowers nor images of parts of flowers. I'm afraid the license will prohibit to use the purchased image as the main eye catching item of a set of cards or clothes you are going to sell. It's your job to read (or to hire someone to read for you) what rights a purchased license gives to you.The same item can have several prices for different sets of rights.
