Q:

How to create and format a partition using a bash script? 

Is there any way to create and format a partition using a bash script? 
I think it can be done with fdisk but I don't know how to feed commands from the bash script into the fdisk shell then exit the fdisk shell. 
I'm wanting to create a partition then format it to ntfs from within bash. 

A:

Similar to the previous suggestions, piping commands to fidsk,  I've found this approach useful to leave details for subsequent maintainers.  The sed bits strip off all the comments before fdisk gets the input. 
# to create the partitions programatically (rather than manually)
# we're going to simulate the manual input to fdisk
# The sed script strips off all the comments so that we can 
# document what we're doing in-line with the actual commands
# Note that a blank line (commented as "defualt" will send a empty
# line terminated with a newline to take the fdisk default. 
sed -e 's/\s*\([\+0-9a-zA-Z]*\).*/\1/' << EOF | fdisk ${TGTDEV}
  o # clear the in memory partition table
  n # new partition
  p # primary partition
  1 # partition number 1
    # default - start at beginning of disk 
  +100M # 100 MB boot parttion
  n # new partition
  p # primary partition
  2 # partion number 2
    # default, start immediately after preceding partition
    # default, extend partition to end of disk
  a # make a partition bootable
  1 # bootable partition is partition 1 -- /dev/sda1
  p # print the in-memory partition table
  w # write the partition table
  q # and we're done
EOF

A:

sfdisk
sfdisk is a Scripted version of fdisk
It is part of util-linux, just like fdisk, so availability should be the same. 
A partition table with a single partition that takes the whole disk can be 
 created with:
echo 'type=83' | sudo sfdisk /dev/sdX

and more complex partition tables are explained below. 
To generate an example script, get the setup of one of your disks:
sudo sfdisk -d /dev/sda > sda.sfdisk

Sample output on my Lenovo T430 Windows 7 / Ubuntu dual boot:
label: dos
label-id: 0x7ddcbf7d
device: /dev/sda
unit: sectors

/dev/sda1 : start=        2048, size=     3072000, type=7, bootable
/dev/sda2 : start=     3074048, size=   195430105, type=7
/dev/sda3 : start=   948099072, size=    28672000, type=7
/dev/sda4 : start=   198504446, size=   749594626, type=5
/dev/sda5 : start=   198504448, size=   618891264, type=83
/dev/sda6 : start=   940277760, size=     7821312, type=82
/dev/sda7 : start=   817397760, size=    61437952, type=83
/dev/sda8 : start=   878837760, size=    61437500, type=83

Once you have the script saved to a file, you can apply it to sdX with:
sudo sfdisk /dev/sdX < sda.sfdisk

For sfdisk input, you can just omit the device names, and use lines of type:
start=        2048, size=     3072000, type=7, bootable

They are just ignored if present, and the device name is taken from the command line argument. 
Some explanations:

header lines: all optional:

label: type of partition table.  dos (MBR) is the old an widely supported one, gpt the new shiny thing. 
unit: only sector is supported.  1 sector usually equals 512 bytes.  Find with cat /sys/block/sda/queue/hw_sector_size See also: https://unix.stackexchange.com/questions/2668/finding-the-sector-size-of-a-partition
device: informative only I think

partition lines:

start: offset inside the disk at which the partition starts. 
start has very good defaults, and can often be ommited:

on the first line, start is 2048, i.e. 1Mb (2048 + 512), which is a sane default for disk compatibility
further start default to the first unallocated position 

size: man sfdisk says: The default value of size indicates "as much as possible".  So to fill the disk with a single partition use: /dev/sda : start=2048, type=83
type: magic byte stored on the boot sector for each partition entry.  Possible values: https://en.wikipedia.org/wiki/Partition_type On this example we observe:

7 (sda1, 2 and 3): filesystems that Windows supports.  Preinstalled Windows stuff and Lenovo recovery partitions.  sudo blkid labels help identify them. 
5 (sda4): extended primary partition, which will contain other logical partitions (because we can only have 4 primary partitions with MBR)
83(sda5, 7, and 8): partitions which Linux supports.  For me one home, and two roots with different Ubuntu versions
82 (sd6): swap

fdisk can also read sfdisk scripts with the I command, which "sources" them during an interactive fdisk session, allowing you further customization before writing the partition. 
Tested on Ubuntu 16.04, sfdisk 2.27.1. 
Format and populate the partitions an image file without sudo
This is a good way to learn to use sfdisk without blowing up your hard disks: https://stackoverflow.com/questions/10949169/how-to-create-a-multi-partition-sd-disk-image-without-root-privileges/52850819#52850819

A:

fdisk reads from stdin so you just need to feed it the appropriate commands.   For example, the following clears the partition table, if there is one, and makes a new one that has a single partition that is the; entire disk:
(
echo o # Create a new empty DOS partition table
echo n # Add a new partition
echo p # Primary partition
echo 1 # Partition number
echo   # First sector (Accept default: 1)
echo   # Last sector (Accept default: varies)
echo w # Write changes
) | sudo fdisk

I recommend you do the task you want, recording what you type so you can reproduce it.