In cases when you can´t use Admin role you can create a custom security role and grant the following permissions:

Companies>Company Maintenance

Inquire - All

Companies>Contacts

Add - All
Edit - All
Inquire - All

Companies>Manage Attachments

Add - All
Inquire - All

Projects>Project Tickets

Add - All
Edit - All
Inquire - All

Service Desk/Service Tickets

Add - All
Edit - All
Inquire - All

If you would like to allow reopening add the following permissions:

Service Desk>Close Service Tickets

Edit - All
Inquire - All

System>API Reports

Add - All
Edit - All

Delete - All
Inquire - All

System>Manage Hosted API

Add - All
Edit - All

Delete - All
Inquire - All

System>Member Maintenance

Inquire - All

System>My Company

Inquire - All

System>Table Setup

Inquire - All

Edit - All
Add - All
Delete - All

System>Table Setup> Customize 

Company / Company Status

General / Manage Hosted API

General / Workflow Rules

Service / Service Board

Service / Priority
Service/ Source

Company / Company Type

Time & Expense>Time Entry

Inquire - All

After that assign the created security role to the integration member.