The Issue and Task data is accessible from the API (both REST and Query) the same way as other record data.

NOTE: Your organization will require Developer Pack to leverage API.

In order to interact with Issues and Tasks via the API, you will first need to get the Form ID for each one.

In order to obtain the Form ID for either Tasks or Issues, utilize Get All Forms and set the type to "system".

On the right hand side, enter your organization's API token as the Header then click "Try It!".

The response will include 2 forms: "Issues" and "Tasks". You can search and copy the "form_id" value for each one that you are looking to interact with. One thing to note is that the search will display 3 "form_id" values. You will notice that two of the values are the same (the Tasks' form id).

Here is a sample response where you can locate the "form_id":

Now that you have the Form ID, you can utilize it with the Records API or Query API.