I'd guess that you are using Avant Garde Gothic. There is a version of Avant Garde Gothic that has the"l-with-stroke", the Paneuropean version, but so far as I know it doesn't have a "modifier-letter-small-w". That glyph isn't used in Europe at all, I think. So it may be that Willi is correct (the website is substituting a different font for those glyphs that Avant Garde Gothic doesn't have) or perhaps that your browser is substituting glyphs. The l-with-stroke in your first screenshot looks nothing at all like the one in Avant Garde Gothic Paneuropean; it looks like Arial Unicode MS to me, which is the font that your browser would most likely be used to subsitute the small-w modifier, assuming you're using Windows here. Pretty sure you are, because a Mac or a phone would most likely use Noto, and that modifier w looks nothing at all like the one in Noto.
I've also seen rare cases (okay, one case) where the World Ready Composer was the thing that would make the modifier-small-w render correctly. The WRC actually is used in lots of left-to-right languages like Khmer and Burmese and Lao, and which in those languages actually can cause glyphs to hide, or be transformed into accents or in other ways. But you're not posting in an academic linguistics mailing list about your homemade OT font that you made with no comprehension of typography, so that's probably not the case for you. 😉 You just need to use a font that has those glyphs for rendering that Native language. Arial Unicode MS seems like the obvious choice here, but there are plenty of fonts that look much better than have good support for these languages.
But I'm guessing that you'd rather match your organization's preexisting graphic identity stuff, and so want to stick with Avant Garde. If you're using paragraph styles here, it's quite easy to set up a GREP Style that will capture only the glyphs that aren't in your font, and set them in a character style with a font that does support them. If it's just a few dropped glyphs, then it might be faster to use the Find/Change dialog, or even to do it manually.
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