Ok,
so lets talk through the troubleshooting to help 

So, the loop  for(OrderApi__Subscription__c sub: Trigger.new) works, but not  for(OrderApi__Subscription__c sub:subscription){

so subscription is the change
where do we get subscription:

List<OrderApi__Subscription__c> subscription = [SELECT ID, OrderApi__Term_End_Date__c FROM OrderApi__Subscription__c WHERE id IN: listIds AND OrderApi__Term_End_Date__c != NULL Order By OrderApi__Term_End_Date__c desc LIMIT 1];

As we look at the query we see WHERE id IN: listIds

where does listIds come from:

for (OrderApi__Subscription__c childObj : Trigger.new) {
    listIds.add(childObj.OrderApi__Account__c);
    }

SO listIds is a list of (i presume) account Ids.

lets look at the query - FROM OrderApi__Subscription__c WHERE id IN: listIds - so we are trying to pull OrderApi__Subscription__c records where their ID matches an account Id.

so lets fix some lines:

trigger updateAccount on OrderApi__Subscription__c (after insert, after update) {
 
    Map<ID, Account> parentAcc;
//lets be clear about what list is what
//    List<Id> listIds = new List<Id>();
List<Id> accIds = new List<Id>();
List<Id> oasIds = new List<Id>();
    List<Account> accountList = new List<Account>();
    List<Account> accountList1 = new List<Account>();
        
for (OrderApi__Subscription__c childObj : Trigger.new) {
//  listIds.add(childObj.OrderApi__Account__c);
    accIds.add(childObj.OrderApi__Account__c);
    oasIds.add(childObj.Id);
    }
    
 //   Integer soql = [Select count() FROM OrderApi__Subscription__c WHERE OrderApi__Account__r.Id =:listIds AND OrderApi__Is_Active__c != FALSE];

    Integer soql = [Select count() FROM OrderApi__Subscription__c WHERE OrderApi__Account__c =:accIds AND OrderApi__Is_Active__c != FALSE];

    parentAcc = new Map<Id, Account>([SELECT id, Collegiate_Current_Term_Start_Date__c, Member_Count__c, Collegiate_Paid_Through_Date__c, Collegiate_Current_Term_End_Date__c, Collegiate_Chapter_Status__c, Collegiate_Chapter_Activated_Date__c  FROM Account WHERE ID IN :listIds]);
    
//    List<OrderApi__Subscription__c> subscription = [SELECT ID, OrderApi__Term_End_Date__c FROM OrderApi__Subscription__c WHERE id IN: listIds AND OrderApi__Term_End_Date__c != NULL Order By OrderApi__Term_End_Date__c desc LIMIT 1];

    List<OrderApi__Subscription__c> subscription = [SELECT ID, OrderApi__Term_End_Date__c FROM OrderApi__Subscription__c WHERE id IN: oasIds AND OrderApi__Term_End_Date__c != NULL Order By OrderApi__Term_End_Date__c desc LIMIT 1];

    Subscription__c mhc = Subscription__c.getInstance();
    string ItemCollID = mhc.Collegiate_Item_Id__c;
    
    for(OrderApi__Subscription__c sub:subscription){
       if(sub.OrderApi__Item__c != NULL && sub.OrderApi__Item__c == ItemCollID) {
            Account newAcc = parentAcc.get(sub.OrderApi__Account__c);
            newAcc.Collegiate_Chapter_Status__c=sub.OrderApi__Status__c;
            newAcc.Collegiate_Chapter_Activated_Date__c=sub.OrderApi__Activated_Date__c;
            newAcc.Collegiate_Paid_Through_Date__c =sub.OrderApi__Paid_Through_Date__c;
            newAcc.Collegiate_Current_Term_End_Date__c=sub.OrderApi__Current_Term_End_Date__c;
            newAcc.Collegiate_Current_Term_Start_Date__c=sub.OrderApi__Current_Term_Start_Date__c;
            accountList.add(newAcc);
        }
    }
    
    for(OrderApi__Subscription__c subs:Trigger.new){
        Account newAcc1 = parentAcc.get(subs.OrderApi__Account__c);
        newAcc1.Member_Count__c = soql;
        accountList1.add(newAcc1);
    }
    if(accountList.size()>0) {
       update accountList;
    }
    if(accountList1.size()>0) {
       update accountList1;
    }     
}

Thing to take away from this - use clear variable names - listIds as a variable name is not clear enough (in my mind) but accIds or oasIds is clearer.
You could go accountIds or AccountIdsList or similar, but they do start to become unweildy - example - listOrderApiSubscriptionIds

HTH

Regards
Andrew