Hi,

By default the button is visible for all the users. If you check the code of the button you can see a groups attribute along with the button in it the given group is base.group_user, so that it will be visible for all the internal users.

You can inherit the button and change the group to the group of the inventory manager or edit and change the group from the front end.

To change it from the front end activate the developer mode and click the debugger button and click the Edit Form view and inside the Architecture page of the opening wizard you can see the button Check Availability with the name action_assign , here change the groups to stock.group_stock_manager.

If you are looking to change the same by a custom module by inheriting you can check the below code:

<record id="picking_form_inherit" model="ir.ui.view">
    <field name="name">contacts.custom.fields</field>        
    <field name="model">stock.picking</field>        
    <field name="type">form</field>        
    <field name="inherit_id" ref="stock.view_picking_form"/>        
    <field name="arch" type="xml">
<xpath expr="//button[@name='action_assign']" position="attributes">
    <attribute name="groups">stock.group_stock_manager</attribute>
</xpath>
</field>
</record>

Thanks