I figured out a way to do it. Part of the custom activity in ADF v1 is the Execute method that has a context parameter. From that context you can get the connection string to the blob storage and the path of the blob and then you can extract the sas token like this:
public override IDictionary<string, string> Execute(
AOMDotNetActivityContext context,
IActivityLogger logger)
{
string blobConnectionString = context.ConnectionString;
CloudStorageAccount inputStorageAccount = CloudStorageAccount.Parse(blobConnectionString);
var blob = new CloudBlob(new Uri(inputStorageAccount.BlobEndpoint, Path.Combine(context.FolderPath, context.FileName)), inputStorageAccount.Credentials);
SharedAccessBlobPolicy adHocSAS = new SharedAccessBlobPolicy()
{
SharedAccessExpiryTime = DateTime.UtcNow.AddHours(48),
Permissions = SharedAccessBlobPermissions.Read | SharedAccessBlobPermissions.Delete
};
string sasBlobToken = blob.GetSharedAccessSignature(adHocSAS);
string fullUri = new Uri(blob.Uri, sasBlobToken).ToString();