Does Chinese require more context to avoid ambiguities than English?
TL;DR I would say so, yes.
One type of ambiguity is given by the isolating nature of the language (i.e. absence of inflection) combined with a high frequency of homophones.
Which means if you hear me saying bei1 ju4, am I talking about a tragedy 悲剧 or glassware 杯具？This requires context to disambiguate.
You might think: &quot;well that's easy to disambiguate&quot;. Okay, what about you hear me saying: 我的 ji4 yi4 很差! Is it 记忆 (memory) or 技艺 (skill)?!
As you can see, this ambiguity exists in speech only. In writing, you readily identify what I'm talking about by just looking at the actual characters.

Another type of ambiguity is given by the lack of verb tenses, which makes it harder to figure out if someone is talking about a past, present or future event. This usually requires time indicators (那时候，今天，明天，去年，当时), particles or auxiliaries (过，了，将，要), or a larger frame of reference, e.g. a previous sentence.
But it's possible to argue, and it's a valid argument in my opinion, that this kind of ambiguity practically exists only when you compare Chinese to a language with different grammatical features, and not in absolute. As your proficiency level increases, you simply get used to how the language flows.
And at that point the comparison with English is maybe a little moot: the two languages are just 不一样。

Edit:
As an additional note, English itself doesn't lack weird ambiguities, apparently known as garden-path sentences