The use of thin membranes to suspend an inertial fusion energy fuel capsule in a holder or hohlraum for injection into a reaction chamber is investigated. Also discussed is the stress that occurs in the fuel within a capsule during acceleration. To determine the maximum target acceleration, capsule displacement and membrane deformation angle are calculated for an axisymmetric geometry for a range of membrane strain and capsule size. Membranes must be thin (perhaps < 1 µm) to minimize their effect on capsule implosion symmetry. Typical target injection scenarios prefer accelerations in excess of 1000 m/s2. Acceleration in excess of 1600 m/s2 for a 2.4-mm-radius 30-mg capsule is possible with two 0.1-µm-thick membranes. Added stress from vibrations could cause a factor of 2 decrease in the allowed acceleration unless the acceleration profile is modified to mitigate this effect. However, if the acceleration is gradually increased and then decreased, over a few membrane oscillation periods (i.e., a few milliseconds), the membrane stress due to oscillation overshoot and the final capsule oscillation amplitude is minimal. Compared with a single membrane, a dual membrane geometry allows several times greater acceleration with reduced capsule displacement.