Yess of course that's possible. ShinyR provides a function called radioButtons. 
syntax:
radioButtons(inputId, label, choices, selected = NULL, inline = FALSE,
  width = NULL)
Example:
ui <- fluidPage(
 radioButtons("dist", "Distribution type:", c("Normal" = "norm", "Uniform" = "unif", "Log-normal" = "lnorm", "Exponential" = "exp")), 
plotOutput("distPlot") 
)
server <- function(input, output) { 
output$distPlot <- renderPlot({ dist <- switch(input$dist, norm = rnorm, unif = runif, lnorm = rlnorm, exp = rexp, rnorm) 
hist(dist(500)) 
}) 
}