This applies to SoftMaker Office in general, although I have currently only use for the feature in TextMaker.
When I copy&paste fragments of HTML-content from a browser into TextMaker, <img/> is honored in the way that the picture is included in the TextMaker version of the document. The picture appears to be embedded, it has obviously been found in the cache (I hope so) upon pasting into TextMaker.
My suggestion is the same or similar behavior upon opening a html page from hard-disk. My example is the weather-forecast on meteociel: https://www.meteociel.fr/previsions/48191/nurnberg.htm. I do not know how you can ensure the presence of a picture file, but my imagination gives me several solutions. In case of doubt, ask the user to provide a path or something... this is not a big problem and not worth discussing.
When I copy&paste the table to textmaker, the display is immaculate: But as I rather download the weather information with a script, the raw HTML snippet containing the <table/> is stored in a temporary HTML-file. The icons in the outer right column of the table are referenced like this:
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<img alt="Couvert" title="Couvert" src="//static.meteociel.fr/prevision/test/picto/nuageux.gif">
Anyway, TextMaker should show the alternate-text for pictures, i.e. the value of the alt-attribute (in the example above: « Couvert »), each time that it cannot honor the <img/> tag for whichever reason. This is probably even more important than the display of images. In my example of weather data from meteociel, this concerns mainly the wind-directions which are only roughly indicated by the icons, but precise in the alt- and title-attributes.
Edits: Improved wording in an attempt to gain intelligibility.