How to add data to dropdownlist at runtime and rebind in dropdownlist automatically

I agree with what F-ES Sitecore. What you can do is to add an optione "Others" in your DropDownList and once that option is selected, then you can place a TextBox and a Button in a Panel/Div to group them and then trigger that Div to be shown in the page. You have two options to do that if you are working on ASP.NET WebForms.

First option is using Server-side implementation, and that is to handle the DropDownList_SelectedIndexChanged event and on that event, you can check for the "Others" value and show your TextBox to the page where users can input data and then at Button_Click event, you can write a code to insert that data into your database. You don't need a Web API to implement this not unless if your data is coming from different server.

The second option is using Client-side implementation, could be plain JavaScript or jQuery. And you need to create a Web API or Page Method or Web Service to access your data via AJAX call. Here's a quick example using plain JavaScript:

HTML:

<select name="color" onchange='checkvalue(this.value)'> 
    <option>Select Authors</option>  
    <option value="1">You</option>
    <option value="2">Me</option>
    <option value="others">Others</option>
</select>
<div id="divNewEntry" style='display:none'> 
        <input type="text" id="txtNewAuthor" />
        <input type="button" id="btnSave" value="Save" /> 
</div>

JavaScript:

function checkvalue(val)
{
    if(val==="others")
       document.getElementById('divNewEntry').style.display='block';
    else
       document.getElementById('divNewEntry').style.display='none'; 
}

Once you've done that, you can then use jQuery AJAX to communicate with your server (API) to insert the database at Button's Click event something like this:

 $(document).ready(function () {  
            $('#btnSave').click(function () {  
                $.ajax({  
                    type: 'GET',  
                    url: 'api/Author',  
                    dataType: 'json',  
                    success: function (data) {  
  			// You can repopulate your dropdownlist here to reflect the newly added entry
			// then hide the Div here containing the TextBox and Button
                    }  
  
                });  
            });  
 
});  

Here's a great article that talks about how to access your data using jQuery AJAX: Many ways to communicate with your database using jQuery AJAX and ASP.NET[^]