Others have explained why you can't simply omit the "that".
However, in this case it would be idiomatic to omit "
that is", leaving "There is so much at stake for many."
ell.stackexchange.com/users/37000/Especially Lime
No, the relative pronoun that cannot be omitted in the sentence "There is so much (that) is at stake for many" .
This is because that functions as the subject of the defining relative clause that is at stake .
When the relative pronoun is the subject of the relative clause it cannot be omitted.
The same applies to the other relative pronouns.
For example: There are so many people who are in jobs they dislike.
There is so much which is unknown about the universe.
If, on the other hand, the pronoun functions as the object of the relative clause, then it can be omitted: There is so much (that) you can do to make the world a better place.
There are so many people (who/m)
I would like to thank.
There is so much which we don't know about the universe.
Object pronoun omission is very common in spoken English.
More information:
Defining relative clauses
Addendum:
This addendum picks up on points made by @BillJ and @Snailplane
that that in sentences such as the OP's is not regarded as a relative pronoun by some modern linguists.
In chapter 12, section 3.5.6 of
The Cambridge Grammar of the English Language Huddleston, Pullum and Peterson make the case for that being treated as a subordinator, not as a relative pronoun.
They discuss several ways in which that is different from the 'uncontroversial' relative pronouns ( who, which, whose ), including Lack of upward percolation , Finiteness and Omissibility .
Huddleston and Pullum's A Student's Introduction to English Grammar contains a shorter analysis of the same issue.
It contrasts the two sentences:
They rejected the suggestion which your son made.
(relative clause)
They rejected the suggestion that your son was lying.
(content clause)
Clearly, it would be problematic to call that in the second sentence above a relative pronoun, not least because it cannot be replaced by the relative pronoun which .
But, for me the issue is less clear cut in sentences such as:
They rejected the suggestion that your son made.
The Hyundai Santa Fe was the first car that crossed Antarctica.
where that can indeed by replaced by which .
Ultimately, however, I suspect that visitors to this ELL site are less interested in terminology than grammaticality.
The terminology issue would be a good one for the sister site ELU .
There is a discussion of
that as a relativizer on Wikipedia's English relative clauses .
ell.stackexchange.com/users/1296/Shoe
There is so much that ___ is at stake for many Initial point: "that" is not a relative pronoun; it is a clause subordinator.
In this case it is not omissible.
The relativised element is covert; it’s missing and represented by the '___' notation, called 'gap'.
"So much" is antecedent to the gap functioning as Subject, which is the reason that "that" is not omissible.
We understand the relative clause to mean: "so much is at stake for many".
ell.stackexchange.com/users/31780/BillJ
Yes, but you must also omit "is."
"There is so much at stake for many.
"
Now it's a nice, concise phrase.
This kind of elision is common in spoken and written English.
"That is...
" phrases can help clarify and position ideas in complex sentences, but they can often be left out to make shorter, more forceful sentences.
ell.stackexchange.com/users/28413/user8356
When the relative pronoun acts is a clause subordinator and is the subject of a relative clause, it has to be included.
You cannot omit it in this case.
More examples:
I have a robot that can speak.
Mary has got toys that are made of metal.
BBC Learning English
ell.stackexchange.com/users/44134/SovereignSun
There is so much ( that ) is at stake for many.   Can we omit ' that ' in this sentence?
Omitting 'that' in this sentence
ell.stackexchange.com
<relative-pronouns><ellipsis><relative-clauses><subjects>
ell.stackexchange.com/questions/126806
ell.stackexchange.com/users/11976/whitecap
No, the relative pronoun that cannot be omitted in this sentence. When the relative pronoun is the subject of the relative clause it cannot be omitted. When the relative pronoun is the subject of the relative clause it cannot be omitted. 
Relative pronoun that cannot be omitted in the sentence "There is so much (that) is at stake for many" can't be omitted when it is the subject of a relative clause. The same applies to the other relative pronouns. When the relative pronoun is the subject of the relative clause it cannot be omitted, and so happens to the rest of the pronouns as well