In one of my projects I used SQL Server Compact 4.0 and my connection string was simple enough:
<connectionStrings>
<add name="MyDB"
connectionString="Data Source=|DataDirectory|MyDB.sdf"
providerName="System.Data.SqlServerCe.4.0" />
</connectionStrings>
I think that you should check your providerName. I am not sure if System.Data.SqlClient is the right one.
Have you installed EntityFramework.SqlServerCompact NuGet package? Once it is installed System.Data.SqlServerCe.4.0 provider name is added and it should be used in the connection string.
Check if SqlServerCompact provider is added to your web.config.
<entityFramework>
<defaultConnectionFactory type="System.Data.Entity.Infrastructure.SqlCeConnectionFactory, EntityFramework">
<parameters>
<parameter value="System.Data.SqlServerCe.4.0" />
</parameters>
</defaultConnectionFactory>
<providers>
<provider invariantName="System.Data.SqlClient" type="System.Data.Entity.SqlServer.SqlProviderServices, EntityFramework.SqlServer" />
<provider invariantName="System.Data.SqlServerCe.4.0" type="System.Data.Entity.SqlServerCompact.SqlCeProviderServices, EntityFramework.SqlServerCompact" />
</providers>
</entityFramework>
<system.data>
<DbProviderFactories>
<remove invariant="System.Data.SqlServerCe.4.0" />
<add name="Microsoft SQL Server Compact Data Provider 4.0" invariant="System.Data.SqlServerCe.4.0" description=".NET Framework Data Provider for Microsoft SQL Server Compact" type="System.Data.SqlServerCe.SqlCeProviderFactory, System.Data.SqlServerCe, Version=4.0.0.0, Culture=neutral, PublicKeyToken=89845dcd8080cc91" />
</DbProviderFactories>
</system.data>
Data Source=MyData.sdf;Persist Security Info=False;(noAttachDbFileName=attribute or anything of that sort....)