openUtility() can be used to launch Salesforce Omni-Channel widget on click of a button.

Sample Lighting Aura Component Code:

Component:

<aura:component implements="flexipage:availableForAllPageTypes">
    <lightning:utilityBarAPI aura:id="utilitybar"/>
    <lightning:card>
        <lightning:button 
                          variant="brand" 
                          label="Launch Omni-Channel Widget" 
                          onclick="{! c.launchOmni }">
        </lightning:button>
    </lightning:card>
</aura:component>

JavaScript:

({
    launchOmni : function( component, event, helper ) {
        
        let utilityAPI = component.find( "utilitybar" );
        utilityAPI.getAllUtilityInfo().then(function( response ) {
            /*
             * Following Code will open the first Utility. 
             * Add Omni-Channel as the first Utility.
            */
            let myUtilityInfo = response[ 0 ];
            utilityAPI.openUtility( {
                utilityId: myUtilityInfo.id
            } );
       })
        .catch(function( error ) {
            console.log( 'Error occurred', JSON.stringify( error ) );
        });
        
    }, 
    
})

Output: