you should change something like below:
best practice: 
// sample in template template:
<input type="text" class="form-control form-control-sm" v-bind:value=meminfo.last_name name="online_membership[last_name]" :ref="online_membership[last_name]" >
Script:
methods() {
   function() {
     // i would like to access the input value of this ref
     this.$refs[online_membership[last_name]]
   }
}
but still if you want to use as a string including '[' then simply you can use below solution:
// sample in template template:
<input type="text" class="form-control form-control-sm" v-bind:value=meminfo.last_name name="online_membership[last_name]" ref="'online_membership[last_name]'" >
Script:
methods() {
   function() {
     // i would like to access the input value of this ref
     this.$refs['online_membership[last_name]']
   }
}
hope it will solve your problem