My property management company uses Podio to store property data.
I want the ability to send that property data directly into my WordPress website. Essentially, once all the data is in, I want it to be pushed directly to the website.
Are API hooks the most effective way to do this? Or is there a better way?
Globiflow would be an interesting candidate for you to consider.
They offer the ability to publish information from your app to a CMS page. http://www.globiflow.com/help/external-cms-page.php
Alternatively, if you have strong development skills, you could use the Create data feed option, to publish your app (or a view in your app) to a JSON/XML feed. http://www.globiflow.com/help/creating-data-feeds.php
In both cases, the information is automatically pushed when you create new items.