If you're talking about alternative text specifically on web pages (alt text), since alt is an attribute of the HTML img tag it's part of web page code, not carried with image files. The metadata fields in Lightroom are based on the IPTC standard, which doesn't have an alt text field as far as I know, so Lightroom (and other image organizer applications) don't have a field specifically for alt text. There's no direct connection between IPTC metadata and HTML because the IPTC metadata standard evolved independently of and before the web.
But because alt text is so important for accessibility, some websites can convert one of the provided IPTC fields into alt text. For example, PhotoShelter creates alt tag text from the IPTC Description field (caption) of each uploaded photo. If you were building a PhotoShelter website, in Lightroom you'd make sure the text you enter in the Description/Caption field would make good alt tags.
So the answer to the question is, it depends on what software runs the web sites on which you need alt text for images. If the software has a way to automatically take text from a metadata field and use it as alt text, find out which field or fields it can draw from for that purpose and then enter the alt text into that field in Lightroom. Then, when you use the Export dialog box in Lightroom, be sure to choose a Metadata option that does not strip that field on the way out.
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As far as I know, Lightroom cannot add alt text to an image.
Alt text is not metadata, but an attribute for the image tag in html.
So to add alt text to images on a webpage, you have to use a program like Dreamweaver to edit the html - any text editor will also do the job.
The html for an image with the alt attribute will look something like this:
<img src="images/image1.jpg" alt="Alt text goes here"/>