[release/8.0] Update dependencies from dotnet/emsdk (#120900)

This pull request updates the following dependencies

[marker]: <> (Begin:4ebef09c-22a4-4345-9e95-08db9f47cad7)
## From https://github.com/dotnet/emsdk
- **Subscription**:
[4ebef09c-22a4-4345-9e95-08db9f47cad7](https://maestro.dot.net/subscriptions?search=4ebef09c-22a4-4345-9e95-08db9f47cad7)
- **Build**:
[20251027.3](https://dev.azure.com/dnceng/internal/_build/results?buildId=2825390)
([288696](https://maestro.dot.net/channel/3073/github:dotnet:emsdk/build/288696))
- **Date Produced**: October 27, 2025 12:50:48 PM UTC
- **Commit**:
[81d2833ee1bf80f65cae9957c1926993ce1a299f](dotnet/emsdk@81d2833)
- **Branch**:
[release/8.0](https://github.com/dotnet/emsdk/tree/release/8.0)

[DependencyUpdate]: <> (Begin)

- **Updates**:
  - From [8.0.22-servicing.25516.3 to 8.0.22-servicing.25527.3][5]
     - Microsoft.SourceBuild.Intermediate.emsdk
  - From [8.0.22 to 8.0.22][5]
     - Microsoft.NET.Workload.Emscripten.Current.Manifest-8.0.100

[5]: dotnet/emsdk@9e37ff5...81d2833

[DependencyUpdate]: <> (End)


[marker]: <> (End:4ebef09c-22a4-4345-9e95-08db9f47cad7)

---------

Co-authored-by: dotnet-maestro[bot] <dotnet-maestro[bot]@users.noreply.github.com>

Author: dotnet-maestro[bot]

NuGet.config

@@ -9,7 +9,7 @@
     <clear />
     <!--Begin: Package sources managed by Dependency Flow automation. Do not edit the sources below.-->
     <!--  Begin: Package sources from dotnet-emsdk -->
-    <add key="darc-pub-dotnet-emsdk-9e37ff5" value="https://pkgs.dev.azure.com/dnceng/public/_packaging/darc-pub-dotnet-emsdk-9e37ff5e/nuget/v3/index.json" />
+    <add key="darc-pub-dotnet-emsdk-81d2833" value="https://pkgs.dev.azure.com/dnceng/public/_packaging/darc-pub-dotnet-emsdk-81d2833e/nuget/v3/index.json" />
     <!--  End: Package sources from dotnet-emsdk -->
     <!--  Begin: Package sources from dotnet-sdk -->
     <!--  End: Package sources from dotnet-sdk -->

eng/Version.Details.xml

@@ -92,11 +92,11 @@
     </Dependency>
     <Dependency Name="Microsoft.NET.Workload.Emscripten.Current.Manifest-8.0.100" Version="8.0.22">
       <Uri>https://github.com/dotnet/emsdk</Uri>
-      <Sha>9e37ff5ebf5f464d80bdae6ad9d24e7a01ee11f8</Sha>
+      <Sha>81d2833ee1bf80f65cae9957c1926993ce1a299f</Sha>
     </Dependency>
-    <Dependency Name="Microsoft.SourceBuild.Intermediate.emsdk" Version="8.0.22-servicing.25516.3">
+    <Dependency Name="Microsoft.SourceBuild.Intermediate.emsdk" Version="8.0.22-servicing.25527.3">
       <Uri>https://github.com/dotnet/emsdk</Uri>
-      <Sha>9e37ff5ebf5f464d80bdae6ad9d24e7a01ee11f8</Sha>
+      <Sha>81d2833ee1bf80f65cae9957c1926993ce1a299f</Sha>
       <SourceBuild RepoName="emsdk" ManagedOnly="true" />
     </Dependency>
     <Dependency Name="Microsoft.SourceBuild.Intermediate.source-build-reference-packages" Version="8.0.0-alpha.1.25462.3">