As Jesse Carter mentioned that we could use Kudu Zip Api to do that. I do a demo for that. It works correctly on my side. The following is my detail steps:
Preparation:
Registry an AD application and assign role to applcation, more details please refer to Azure official tutorials. After that we can get tenantId, appId, secretKey from the Azure Portal.
1.Prepare an authentication file, we could get more information from github document.
subscription=########-####-####-####-############
client=########-####-####-####-############
tenant=########-####-####-####-############
managementURI=https\://management.core.windows.net/
baseURL=https\://management.azure.com/
authURL=https\://login.windows.net/
graphURL=https\://graph.windows.net/
2.Zip the need to be published file
Steps:
1.Create a C# console project
2.Reference the Microsoft.Azure.Management.ResourceManager.Fluent and Microsoft.Azure.Management.AppService.Fluent, more detail info please refer to the packages.config file section.
3.Add the following code in the Program.cs file
var credentials = SdkContext.AzureCredentialsFactory.FromFile(@"authentication file path");
var azure = Azure
.Configure()
.WithLogLevel(HttpLoggingDelegatingHandler.Level.Basic)
.Authenticate(credentials)
.WithDefaultSubscription();
var webFunctionAppName = "azure function name";
var webFunctionApp = azure.AppServices.FunctionApps.List().Where(x => x.Name.Equals(webFunctionAppName))?.First();
var ftpUsername = azure.AppServices.FunctionApps.GetById(webFunctionApp.Id).GetPublishingProfile().FtpUsername;
var username = ftpUsername.Split('\\').ToList()[1];
var password = azure.AppServices.FunctionApps.GetById(webFunctionApp.Id).GetPublishingProfile().FtpPassword;
var base64Auth = Convert.ToBase64String(Encoding.Default.GetBytes($"{username}:{password}"));
var file = File.ReadAllBytes(@"zip file path");
MemoryStream stream = new MemoryStream(file);
using (var client = new HttpClient())
{
client.DefaultRequestHeaders.Add("Authorization", "Basic " + base64Auth);
var baseUrl = new Uri($"https://{webFunctionAppName}.scm.azurewebsites.net/");
var requestURl = baseUrl+ "api/zip/site/wwwroot";
var httpContent = new StreamContent(stream);
var response = client.PutAsync(requestURl, httpContent).Result;
}
4.Test from the local

5.Check the published result from the Azure kudu tool(https://yourazurefunctionanme.scm.azurewebsites.net/)

packages.config
<?xml version="1.0" encoding="utf-8"?>
<packages>
<package id="Microsoft.Azure.Management.AppService.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Batch.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Cdn.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Compute.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.ContainerRegistry.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Dns.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.DocumentDB.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Graph.RBAC.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.KeyVault.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Network.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Redis.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.ResourceManager.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.ServiceBus.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Sql.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.Storage.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.Azure.Management.TrafficManager.Fluent" version="1.1.3" targetFramework="net452" />
<package id="Microsoft.IdentityModel.Clients.ActiveDirectory" version="2.28.3" targetFramework="net452" />
<package id="Microsoft.Rest.ClientRuntime" version="2.3.8" targetFramework="net452" />
<package id="Microsoft.Rest.ClientRuntime.Azure" version="3.3.8" targetFramework="net452" />
<package id="Microsoft.Rest.ClientRuntime.Azure.Authentication" version="2.3.0" targetFramework="net452" />
<package id="Newtonsoft.Json" version="9.0.1" targetFramework="net452" />
</packages>