Yes. There are latitude and longitude fields in the widget settings. These will override the Location field if specified. They need to be in decimal format and they will return the closest station to your location.
But, there is no support for a shortcode in Tide Graph and no plan to add that. Instead our new plugin called WorldTides does have support for shortcodes:
WorldTides Plugin
WorldTides is free for 30 days (or 100 predictions) so you can test it. Just create a free account and give it a try. After that, you can pay $9.99 for tens of thousands of predictions which could last you years (depending on your needs). The reason it costs money is because I have to pay for the data myself before I pass it along to you. However, instead of charging hundreds of dollars (like the UKHO does), it costs a fraction of a penny for a prediction. That’s why this is the only service of it’s kind. I barely break even on the deal right now but I hope it will grow.