Fixed the issue, the new code is below. I found out that my BlobServiceClient was not properly defined.
const sasURL = "https://myaccountname.blob.core.windows.net/" + "?" + accessKey;
const blobServiceClient = new BlobServiceClient(sasURL);
const containerClient = blobServiceClient.getContainerClient(container);
const blobName = parts[0].filename;
const blockBlobClient = containerClient.getBlockBlobClient(blobName);
const uploadBlobReponse = await blockBlobClient.upload(parts[0].data, parts[0].data.length);