You need to execute the query in your constructor to get the data to show up when the page loads initially. You need to add a reRender attribute to the search command button to get the new list to display when that button is clicked. Also, the extension name in your visualforce page does not seem to match the actual controller name, and the soql queries are not querying the same object as your list definition. Additionally, when you are using dynamic soql, as you are here, you cannot bind variables the way you are doing it. You have to concatenate the strings, and make sure that they are enclosed in single quotes. Here's your code, with the additions in bold:

<apex:page standardController="customobject__c" extensions="FARsearchpageController">
<apex:form >
<apex:outputPanel id="errorPanel">
        <apex:pageMessages />
    </apex:outputPanel>

   <apex:pageblock title="Searach Records" >
 
       <apex:outputText value="Name"/>&nbsp;&nbsp;&nbsp;
       <apex:inputtext value="{!inputNametext}"/> &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
     
       <apex:outputText value="Account Name"/>
       <apex:inputtext value="{!inputAccounttext}"/> &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
     
       <apex:outputtext value="Contact Name"/>
       <apex:inputtext value="{!inputContacttext}"/> &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;
     
       <apex:commandbutton value=" Search " action="{!searchRecords}" reRender="accListBlock" />
   </apex:pageblock>
   <apex:pageBlock id="accListBlock">
     <apex:pageblocktable value="{!accList}" var="acc">
      <apex:column >
         <apex:outputLink value="/{!acc.id}"><apex:outputText value="{!acc.Name}" />
         </apex:outputLink>
         </apex:column>
      <apex:column value="{!acc.Account__c}"/>
      <apex:column value="{!acc.Contact__c}"/>
     </apex:pageblocktable>   
     </apex:pageBlock>
</apex:form>
</apex:page>

Apex class:

Public class  FARsearchpageController {
   Public string inputNametext{get;set;}
   Public string inputAccounttext{get;set;}
   Public string inputContacttext{get;set;}
   Public string inputstatustext{get;set;}
   Public String soql{get;set;}
   Public List<Fuel_Analysis_Report__c> accList{get;set;}
   Public String usernames=UserInfo.getName();

   public FARsearchpageController (ApexPages.StandardController controller) {
       String soql = 'select Name,Account__c,ContactStatus__c,Closed_Date__c  from Fuel_Analysis_Report__c where name != null and Name=\''+Usernames+'\'';
       accList = Database.query(soql);
    }
      
   Public void searchRecords(){
     soql = 'select Name,Account,Contact,Status__c,Closed_Date__c from Fuel_Analysis_Report__c where Name =\''+Usernames+'\'';

if (string.isnotBlank(inputAccounttext)){
    soql += ' and Account.Name like '+'\''+'%'+inputaccounttext+'%'+'\'';
    }
    if (string.isnotBlank(inputnametext)){
    soql += ' and Name like '+'\''+'%'+inputnametext+'%'+'\'';
       }
    if (string.isnotBlank(inputContacttext)){
    soql += ' and Contact.Name like '+'\''+'%'+inputcontacttext+'%'+'\'';
       }
    accList= Database.query(soql);
    }
  }
}
}