Yes it is possible, by writing a custom model binder.  However as GregH says in the comments it may be simpler to split the string at new lines only where required at the back end...
If a textarea is suitable and convenient from a UI perspective, that's great.  In that case I would argue that you stick with default model binder and only convert to List<string> when required (such as db insert).
Otherwise a dynamic list of inputs is possible with some javascript - an add input button.  These 1-to-n input elements will be picked up by the default model binder in to a List<string> automatically just by using the same attribute name on the input element.  See Model binding to a list
There are only two html elements of relevance here: textarea and input.  The difference between them is that a textarea natively allows for multi-line (text-wrapping and/or line breaks) whereas input must not.  Pressing enter in a text area creates a new line whereas pressing enter in an input submits the entire form.