A user must have the Manage Web site and Create Subsites permissions
to add an app for SharePoint. By default, these permissions are
available only to users who have the Full Control permission level or
who are in the site Owners group.
This (I believe) is because the installation of an App implies in the creation of an APP Web - a Subsite of where the app is being installed.
Also, as stated by the documentation, a user cannot grant a permission he does not have.
An app for SharePoint requests the permissions that it needs during
installation from the user who is installing it. The developer of an
app must request, through the app manifest file, the permissions that
the particular app needs to be able to run. (Device and web apps that
access SharePoint, but are not installed to SharePoint websites must
be granted permissions at runtime by the user who is executing the
app. For more information, see Get an overview of apps that request
access permission from SharePoint on the fly.) Users can grant only
the permissions that they have. The user must grant all the
permissions that an app requests or not grant any permission.
Selective grants are not possible. (For apps that request permissions
on the fly, only a user with Manage permissions to the SharePoint
resources that the app seeks to access can run the app, even if the
app is asking only for lesser permissions, such as Read.)
Apparently, asking for Read permissions implies asking for:
- View Items
- Open Items
- View Versions
- Create Alerts
- Use Self-Service Site Creation
- View Pages
while asking for Write permissions implies asking for the read-only permissions as well as:
- Add Items
- Edit Items
- Delete Items
- Delete Versions
- Browse Directories
- Edit Personal User Information
- Manage Personal Views
- Add/Remove Personal Web Parts
- Update Personal Web Parts
To answer the question, you can work around adding the users as owners if you give the Manage web and Create Subsite permissions, but that might be granting more permissions then desired to the users.