Sample code:

Apex Class:

public class LightningDualListSearchController {

    @AuraEnabled( cacheable = true )
    public static List < OptionWrapper > fetchAccounts() {
    
        List < OptionWrapper > listAccounts = new List < OptionWrapper >();
        
        for ( Account acc : [ SELECT Id, Name FROM Account LIMIT 10 ] ) {
        
            listAccounts.add( new OptionWrapper( acc.Name, acc.Id ) );
        
        }
        
        return listAccounts;
    
    }

    public class OptionWrapper {

        @AuraEnabled
        public String label;
        @AuraEnabled
        public String value;
        
        public OptionWrapper( String label, String value ) {
        
            this.label = label;
            this.value = value;
        
        }

    }

}

Lightning Web Component:

HTML:

<template>
    <lightning-card>
        <div class="slds-m-around_medium">
            <lightning-input type="text" label="Search Account" onchange={handleSearchChange}></lightning-input><br/><br/>
            <lightning-button label="Search" variant="brand" onclick={handleSearch}></lightning-button><br/><br/>
            <lightning-dual-listbox
                label="Select/Remove Accounts"
                source-label="Available Accounts"
                selected-label="Selected Accounts"
                options={availableAccounts}
                onchange={handleAccountChange}>
            </lightning-dual-listbox>
        </div>
    </lightning-card>
</template>

JavaScript:

import { LightningElement, wire } from 'lwc';
import fetchAccounts from '@salesforce/apex/LightningDualListSearchController.fetchAccounts';

export default class LightningDualListSearch extends LightningElement {

    availableAccounts;
    searchString;
    initialRecords;

    @wire( fetchAccounts )
    wiredRecs( { error, data } ) {

        if ( data ) {

            console.log( 'Records are ' + JSON.stringify( data ) );
            this.availableAccounts = data;
            this.initialRecords = data;

        } else if ( error ) {

            console.log( 'Error ' + JSON.stringify( error ) );

        }
        
    }

    handleAccountChange( event ) {

        const selectedOptionsList = event.detail.value;
        console.log( 'Selected Options are ' + JSON.stringify( selectedOptionsList ) );

    }

    handleSearchChange( event ) {

        this.searchString = event.detail.value;
        console.log( 'Search String is ' + this.searchString );

    }

    handleSearch( event ) {

        console.log( 'Search String is ' + this.searchString );

        if ( this.searchString ) { 

            this.availableAccounts = this.initialRecords;
 
             if ( this.availableAccounts ) {

                let recs = [];
                for ( let rec of this.availableAccounts ) {

                    if ( rec.label.toLowerCase().includes( this.searchString ) ) {

                        recs.push( rec );
               
                    }
                    
                }

                console.log( 'Matched Accounts are ' + JSON.stringify( recs ) );
                this.availableAccounts = recs;

             }
 
        }  else {

            this.availableAccounts = this.initialRecords;

        }        

    }

}

JS-meta.xml:

<?xml version="1.0" encoding="UTF-8"?>
<LightningComponentBundle xmlns="http://soap.sforce.com/2006/04/metadata">
    <apiVersion>52.0</apiVersion>
    <isExposed>true</isExposed>
    <targets>
        <target>lightning__Tab</target>
    </targets>
</LightningComponentBundle>

Output: