In translating the English sentence to Czech, could její be used instead of svoje?
It would make the sentence ambiguous as to whose children she was looking for, but the English sentence includes that ambiguity as well.
Or would její definitely imply that she was looking for someone else's children?
this happens to be one of the czech variants of the who/whom or terminal preposition wars.
no one will fault you for always using the reflexive possessive if the subject is identical to the possessor. and some people will fault you for using the regular possessive even if you have a reason for doing so.
and yes, "její" could be ambiguous, referring to some other "her". or it could mean you are just stressing/contrasting the subject "her".
more than you wanted to know here.
In case you have trouble reading through it, the link svrsheque posted says, among other things, that saying "Hledá její děti" instead of "svoje", is indeed meant to express that she's looking for someone else's children, not her own. And it goes on to say that this rule is not absolute.
In the recent decades, there's been an onslaught of marketing/advertisment texts using "váš" where the grammar would dictate "svůj" - because supposedly it works better psychologically when the advertisement targets you more individually a insists that all these things are YOURS instead of the less striking "svůj" (belonging to you by default, inconspicuously). It's rather obnoxious. And more language-sensitive people frown upon it.
In my opinion, if the subject and the possessor are exactly the same, you should always use "svůj". There are situations where they are not exactly the same, for example "Fandíme našim sportovcům" (We root for our athletes) - the "we" in the subject is not the same as the group that "owns" the athletes (the latter is the whole nation).