Can hypothyroidism lead to fat gain?

Yes, a less active thyroid will reduce metabolic rate and can cause some weight gain. It is not a lot of weight gain though, and a subactive thyroid is not an excuse for obesity (if hypothyroidic, please see a doctor for medication; it burns fat)

A little. Thyroid problems play a negligible role in obesity.

The American Thyroid Association states:

In general, 5-10 pounds of body weight may be attributable to the thyroid, depending on the severity of the hypothyroidism.[1]

Most of the weight gain directly associated with thyroid issues is negligible[2][3]. In fact, the average weight loss from treatment was 4.3kg, most of which was attributed to the excretion of excess body water[4][5]

Pre-existing obesity, a diagnosis of Graves' disease and prior weight loss independently predicted weight gain and weight continued to rise with time. [2]

References

2.^Dale J, Daykin J, Holder R, Sheppard MC, Franklyn JAWeight gain following treatment of hyperthyroidismClin Endocrinol (Oxf).(2001 Aug)
3.^Fox CS, Pencina MJ, D'Agostino RB, Murabito JM, Seely EW, Pearce EN, Vasan RSRelations of thyroid function to body weight: cross-sectional and longitudinal observations in a community-based sampleArch Intern Med.(2008 Mar 24)