The example allows you to upload .RData files. The approach with load and get allows you to assign the loaded data to a variable name of your choice. For the matter of the example being "standalone" I inserted the top section that stores two vectors to your disk in order to load and plot them later.

library(shiny)

# Define two datasets and store them to disk
x <- rnorm(100)
save(x, file = "x.RData")
rm(x)
y <- rnorm(100, mean = 2)
save(y, file = "y.RData")
rm(y)

# Define UI
ui <- shinyUI(fluidPage(
  titlePanel(".RData File Upload Test"),
  mainPanel(
    fileInput("file", label = ""),
    actionButton(inputId="plot","Plot"),
    plotOutput("hist"))
  )
)

# Define server logic
server <- shinyServer(function(input, output) {

  observeEvent(input$plot,{
    if ( is.null(input$file)) return(NULL)
    inFile <- input$file
    file <- inFile$datapath
    # load the file into new environment and get it from there
    e = new.env()
    name <- load(file, envir = e)
    data <- e[[name]]
    
    # Plot the data
    output$hist <- renderPlot({
      hist(data)
    })
  })
})

# Run the application 
shinyApp(ui = ui, server = server)

It is also possible to have an user upload csv's to your Shiny app. The code below shows a small example on how this can be achieved. It also includes a radioButton input so the user can interactively choose the separator to be used.

library(shiny)
library(DT)

# Define UI
ui <- shinyUI(fluidPage(
  
  fileInput('target_upload', 'Choose file to upload',
            accept = c(
              'text/csv',
              'text/comma-separated-values',
              '.csv'
            )),
  radioButtons("separator","Separator: ",choices = c(";",",",":"), selected=";",inline=TRUE),
  DT::dataTableOutput("sample_table")
)
)

# Define server logic
server <- shinyServer(function(input, output) {
  
  df_products_upload <- reactive({
    inFile <- input$target_upload
    if (is.null(inFile))
      return(NULL)
    df <- read.csv(inFile$datapath, header = TRUE,sep = input$separator)
    return(df)
  })
  
  output$sample_table<- DT::renderDataTable({
    df <- df_products_upload()
    DT::datatable(df)
  })
  
}
)

# Run the application 
shinyApp(ui = ui, server = server)