Should Schema.org Organization data about my company be on every page?

A quick point on this. A site I work on recently got a manual action penalty from Google for having organization JSON-LD markup across every single page on their domain. It didn't affect rankings of course but would have impacted all other structured data until resolved. It's one of the minor pitfalls of JSON-LD; that you can put it anywhere and if Google deems it to be too prevalent then it can impede you.

I would only recommend JSON-LD format for structured data in modern SEO, however. Much simpler to implement and Google recommends it too. Just when it comes to http://schema.org/Organization, put it on the homepage.

Google looks for such information in traditional places such as About, Contact, Company Info, the sites header, or the sites footer. This began in the early days of semantics where specific information from any website was collected to better help match search intent with entities, locale, and persons.

The home page is a good place for this, but is not fully traditional. It is okay to have this info in your content anywhere you want if (of course), however, I would make sure your company information can be found in as many of these traditional places as makes sense. If there is space for it, and it would not negatively effect user experience, then I would suggest either the header or footer of each page. This ensures that the information is seen by both users and search engines. This is not a requirement, however, making it easy to find your company contact information ensures more opportunities for success in finding it both by users and search engines.

Company information that can be easily found and understood with or without mark-up is an important element for branding. In fact, branding will not happen without it. You will notice that many major brand sites will have contact information in mark-up in a header or footer. This is likely done for a good reason.

Schema.org mark-up is not a necessity for this, however, highly recommended. In the early days, when search engines relied mostly on early parser models and limited in what data elements could be reliably captured; names, addresses, phone numbers, e-mail addresses, and the like, were focused upon specifically to ensure search reliability. This allowed for fairly accurate capturing of the data which fortunately follows traditional formatting and anomalies are predictable. However, using Schema.org ensures that there is no mistake in passing critical data to search engines. For this, there is an additional reward for it's usage in more places such as the knowledge vault. Again, mark-up may not be required for such places. It just makes the data capture more reliable.