You can only use <apex:inputField>
tags with fields on proper SObjects. When you want to save information to a controller member variable instead you must use one of the more specific alternatives. You can create a picklist of available values and then write it back to your member variable by specifying it as the value of the <apex:selectList>
tag.
<apex:pageBlockSection title="Account Selection">
<apex:selectList value="{!Account_Name_v}">
<apex:selectOptions value="{!options}"/>
</apex:selectList>
</apex:pageBlockSection>
Since this is in a page block section you're probably after standard formatting, in which case you'll want to wrap this input element inside an <apex:pageBlockSectionItem>
tag along with a label like so:
<apex:pageBlockSection title="Account Selection">
<apex:pageBlockSectionItem>
<apex:outputLabel for="acctName"/>
<apex:selectList value="{!Account_Name_v}" id="acctName"
<apex:selectOptions value="{!options}"/>
</apex:selectList>
</apex:pageBlockSectionItem>
</apex:pageBlockSection>
Controller
public List<SelectOption> GetOptions()
{
// You could populate this using a query for accounts
List<SelectOption> options = new List<SelectOption>
{
new SelectOption('acct1', 'Account One'),
new SelectOption('acct2', 'Account Two'),
new SelectOption('acct3', 'Account Three')
};
return options;
}